Lattice Sums Then and Now 9781107039902, 2013002993, 1107039908

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LATTICE SUMS THEN AND NOW

The study of lattice sums began when early investigators wanted to go from mechanical properties of crystals to the properties of the atoms and ions from which they were built (the literature of Madelung’s constant). A parallel literature was built around the optical properties of regular lattices of atoms (initiated by Lord Rayleigh, Lorentz and Lorenz). For over a century many famous scientists and mathematicians have delved into the properties of lattices, sometimes unwittingly duplicating the work of their predecessors. Here, at last, is a comprehensive overview of the substantial body of knowledge that exists on lattice sums and their applications. The authors also provide commentaries on open questions and explain modern techniques that simplify the task of finding new results in this fascinating and ongoing field. Lattice sums in one, two, three, four and higher dimensions are covered.

Encyclopedia of Mathematics and Its Applications This series is devoted to significant topics or themes that have wide application in mathematics or mathematical science and for which a detailed development of the abstract theory is less important than a thorough and concrete exploration of the implications and applications. Books in the Encyclopedia of Mathematics and Its Applications cover their subjects comprehensively. Less important results may be summarized as exercises at the ends of chapters. For technicalities, readers can be referred to the bibliography, which is expected to be comprehensive. As a result, volumes are encyclopedic references or manageable guides to major subjects.

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A. Kushner, V. Lychagin and V. Rubtsov Contact Geometry and Nonlinear Differential Equations L. W. Beineke and R. J. Wilson (eds.) with P. J. Cameron Topics in Algebraic Graph Theory O. J. Staffans Well-Posed Linear Systems J. M. Lewis, S. Lakshmivarahan and S. K. Dhall Dynamic Data Assimilation M. Lothaire Applied Combinatorics on Words A. Markoe Analytic Tomography P. A. Martin Multiple Scattering R. A. Brualdi Combinatorial Matrix Classes J. M. Borwein and J. D. Vanderwerff Convex Functions M.-J. Lai and L. L. Schumaker Spline Functions on Triangulations R. T. Curtis Symmetric Generation of Groups H. Salzmann et al. The Classical Fields S. Peszat and J. Zabczyk Stochastic Partial Differential Equations with Lévy Noise J. Beck Combinatorial Games L. Barreira and Y. Pesin Nonuniform Hyperbolicity D. Z. Arov and H. Dym J-Contractive Matrix Valued Functions and Related Topics R. Glowinski, J.-L. Lions and J. He Exact and Approximate Controllability for Distributed Parameter Systems A. A. Borovkov and K. A. Borovkov Asymptotic Analysis of Random Walks M. Deza and M. Dutour Sikiric Geometry of Chemical Graphs T. Nishiura Absolute Measurable Spaces M. Prest Purity, Spectra and Localisation S. Khrushchev Orthogonal Polynomials and Continued Fractions H. Nagamochi and T. Ibaraki Algorithmic Aspects of Graph Connectivity F. W. King Hilbert Transforms I F. W. King Hilbert Transforms II O. Calin and D.-C. Chang Sub-Riemannian Geometry M. Grabisch et al. Aggregation Functions L. W. Beineke and R. J. Wilson (eds.) with J. L. Gross and T. W. Tucker Topics in Topological Graph Theory J. Berstel, D. Perrin and C. Reutenauer Codes and Automata T. G. Faticoni Modules over Endomorphism Rings H. Morimoto Stochastic Control and Mathematical Modeling G. Schmidt Relational Mathematics P. Kornerup and D. W. Matula Finite Precision Number Systems and Arithmetic Y. Crama and P. L. Hammer (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering V. Berthé and M. Rigo (eds.) Combinatorics, Automata and Number Theory A. Kristály, V. D. Radulescu and C. Varga Variational Principles in Mathematical Physics, Geometry, and Economics J. Berstel and C. Reutenauer Noncommutative Rational Series with Applications B. Courcelle and J. Engelfriet Graph Structure and Monadic Second-Order Logic M. Fiedler Matrices and Graphs in Geometry N. Vakil Real Analysis through Modern Infinitesimals R. B. Paris Hadamard Expansions and Hyperasymptotic Evaluation Y. Crama and P. L. Hammer Boolean Functions A. Arapostathis, V. S. Borkar and M. K. Ghosh Ergodic Control of Diffusion Processes N. Caspard, B. Leclerc and B. Monjardet Finite Ordered Sets D. Z. Arov and H. Dym Bitangential Direct and Inverse Problems for Systems of Integral and Differential Equations G. Dassios Ellipsoidal Harmonics L. W. Beineke and R. J. Wilson (eds.) with O. R. Oellermann Topics in Structural Graph Theory L. Berlyand, A. G. Kolpakov and A. Novikov Introduction to the Network Approximation Method for Materials Modeling M. Baake and U. Grimm Aperiodic Order I J. Borwein et al. Lattice Sums Then and Now R. Schneider Convex Bodies: The Brunn–Minkowski Theory (2nd Edition)

E NCYCLOPEDIA OF M ATHEMATICS AND ITS A PPLICATIONS

Lattice Sums Then and Now J. M. BORWEIN University of Newcastle, New South Wales

M. L. GLASSER Clarkson University, New York

R. C. McPHEDRAN University of Sydney

J. G. WAN Singapore University of Technology and Design

I. J. ZUCKER King’s College London

University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9781107039902 c J. M. Borwein, M. L. Glasser, R. C. McPhedran, J. G. Wan and I. J. Zucker 2013  This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2013 Printed in Great Britain By TJ International Ltd. Padstow Cornwall. A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Borwein, Jonathan M. Lattice sums then and now / J. M. Borwein, University of Newcastle, New South Wales, M. L. Glasser, Clarkson University, R. C. Mcphedran, University of Sydney, J. G. Wan, University of Newcastle, New South Wales, I. J. Zucker, King’s College London. pages cm. – (Encyclopedia of mathematics and its applications ; 150) Includes bibliographical references and index. ISBN 978-1-107-03990-2 (hardback) 1. Lattice theory. 2. Number theory. I. Title. QA171.5.B67 2013 511.3 3–dc23 2013002993 ISBN 978-1-107-03990-2 Hardback Additional resources for this publication at www.carma.newcastle.edu.au / LatticeSums Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Knowledge of lattice sums has been built by many generations of researchers, commencing with Appell, Rayleigh, and Born. Two of the present authorship (MLG and IJZ) attempted the first comprehensive review of the subject 30 years ago. This inspired two more (JMB and RCM) to enter the field, and they have been joined by a member (JGW) of a new generation of enthusiasts in completing this second and greatly expanded compendium. All five authors are certain that lattice sums will continue to be a topic of interest to coming generations of researchers, and that our successors will surely add to and improve on the results described here.

Contents

Foreword by Helaman and Claire Ferguson Preface 1 Lattice sums 1.1 Introduction 1.2 Historical survey 1.3 The theta-function method in the analysis of lattice sums 1.4 Number-theoretic approaches to lattice sums 1.5 Contour integral technique 1.6 Conclusion 1.7 Appendix: Complete elliptic integrals in terms of gamma functions 1.8 Appendix: Watson integrals 1.9 Commentary: Watson integrals 1.10 Commentary: Nearest neighbour distance and the lattice constant 1.11 Commentary: Spanning tree Green’s functions 1.12 Commentary: Gamma function values in terms of elliptic integrals 1.13 Commentary: Integrals of elliptic integrals, and lattice sums References 2 Convergence of lattice sums and Madelung’s constant 2.1 Introduction 2.2 Two dimensions 2.3 Three dimensions 2.4 Integral transformations and analyticity 2.5 Back to two dimensions 2.6 The hexagonal lattice 2.7 Concluding remarks 2.8 Commentary: Improved error estimates

page xi xvii 1 1 2 30 54 63 65 66 67 68 72 72 74 77 79 87 87 89 93 100 104 109 111 112

viii

Contents 2.9 Commentary: Restricted lattice sums 2.10 Commentary: Other representations for the Madelung’s constant 2.11 Commentary: Madelung sums, crystal symmetry, and Debye shielding 2.12 Commentary: Richard Crandall and the Madelung constant for salt References

112 116 117 117 123

3 Angular lattice sums 3.1 Optical properties of coloured glass and lattice sums 3.2 Lattice sums and elliptic functions 3.3 A phase-modulated lattice sum 3.4 Double sums involving Bessel functions 3.5 Distributive lattice sums 3.6 Application of the basic distributive lattice sum 3.7 Cardinal points of angular lattice sums 3.8 Zeros of angular lattice sums 3.9 Commentary: Computational issues of angular lattice sums 3.10 Commentary: Angular lattice sums and the Riemann hypothesis References

125 125 128 131 134 140 142 144 147 149 150 154

4 Use of Dirichlet series with complex characters 4.1 Introduction 4.2 Properties of L-series with real characters 4.3 Properties of L-series with complex characters 4.4 Expressions for displaced lattice sums in closed form for j = 2−10 4.5 Exact solutions of lattice sums involving indefinite quadratic forms 4.6 Commentary: Quadratic forms and closed forms 4.7 Commentary: More on numerical discovery 4.8 Commentary: A Gaussian integer zeta function 4.9 Commentary: Gaussian quadrature References

157 157 157 161

5 Lattice sums and Ramanujan’s modular equations 5.1 Commentary: The modular machine 5.2 Commentary: A cubic theta function identity References

186 197 199 200

6 Closed-form evaluations of three- and four-dimensional sums 6.1 Three-dimensional sums 6.2 Four-dimensional sums

202 202 216

164 175 177 178 179 181 184

Contents 6.3 Commentary: A five-dimensional sum 6.4 Commentary: A functional equation for a three-dimensional sum 6.5 Commentary: Two amusing lattice sum identities References

ix 220 221 223 224

7 Electron sums 7.1 Commentary: Wigner sums as limits 7.2 Commentary: Sums related to the Poisson equation References

226 236 237 245

8 Madelung sums in higher dimensions 8.1 Introduction 8.2 Preliminaries and notation 8.3 A convergence theorem for general regions 8.4 Specific regions 8.5 Some analytic continuations 8.6 Some specific sums 8.7 Direct analysis at s = 1 8.8 Proofs 8.9 Commentary: Alternating series test 8.10 Commentary: Hurwitz zeta function References

247 247 248 249 250 257 259 260 264 289 290 292

9 Seventy years of the Watson integrals 9.1 Introduction 9.2 Solutions for W F (w f ), W F (α f , w f ), and W S (ws ) 9.3 The Watson integrals between 1970 and 2000 9.4 The singly anisotropic simple cubic lattice 9.5 The Green’s function of the simple cubic lattice 9.6 Generalizations and recent manifestations of Watson integrals 9.7 Commentary: Watson integrals and localized vibrations 9.8 Commentary: Variations on W S 9.9 Commentary: Computer algebra References

294 294 299 303 306 310 312 315 316 319 320

Appendix A.1 Tables of modular equations A.2 Character table for Dirichlet L-series A.3 Values of K [N ] for all integer N from 1 to 100

324 325 330 331

Bibliography Index

350 364

Foreword by Helaman and Claire Ferguson

The Borwein Award: ‘Salt’, the sculpture, created in 2004 As sculptor, and also the inventor of the PSLQ integer relations algorithm, I described to the Canadian Mathematical Society the sculpture expressing the Madelung constant μ as follows: μ :=

 n,m, p



(−1)n+m+ p n 2 + m 2 + p2

.

This polished solid silicon bronze sculpture is inspired by the work of David Borwein, his sons and colleagues, on the conditional series above for salt, Madelung’s constant. This series can be summed to give uncountably many constants; one is Madelung’s constant for sodium chloride. This constant is a period of an elliptic curve, a real surface in four dimensions. There are uncountably many ways to imagine that surface in three dimensions; one has negative Gaussian curvature and is the tangible form of this sculpture.

I will now explain some of the creative processes which led to this sculpture. Actually, the inscription on the sculpture reads ‘created in 2004’ but, in the spirit of this book Lattice Sums Then and Now, the creation started much earlier. There are a couple of questions. First: why would a sculptor create a sculpture about NaCl, as in ‘please pass the “nakkle” ’, sodium chloride or salt, a lifeessential mineral? Second: why would a sculptor be interested in Madelung’s constant, a conditionally convergent series, subject to special summability, giving the electrostatic potential of the interpenetrating lattices of sodium (Na+ ) and chlorine (Cl− ) ions? The equals sign in the above equation is misleading at this stage because the right-hand side is not defined as it stands. In fact, the right-hand side is a conditional series of infinitely many positive and negative terms which can be

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Foreword

rearranged to give any real number whatsoever (!); the commutative law holds for finitely many summands but does not hold for infinitely many summands. I will answer these two questions raised above in order. First answer: I was born in the Humbolt Basin in the Rocky Mountains and spent the first five years of my life there. This basin contains the Great Salt Lake and huge areas of evaporated deposits of salt minerals. At age three I saw my natural mother killed by lightning and my natural father drafted into the Pacific theatre of World War II. Between ages three and five I was the ‘guest’ of a large extended family of aunts and uncles. After age five I was adopted by a carpenter and stone mason who lived in upstate New York. There I learned to work with my hands. I was a strange little grass orphan. The aunts wanted to mother me but the uncles had the pragmatic upper hand. How strange was I? One aunt, in particular, recalled that she came in the kitchen and found me at the kitchen table intent on sorting grains of salt. I had at that age some sort of microscopic vision; some of my own children told me they went through a sort of microscopic vision stage and later lost it, as did I. Those little cubettes of salt had a great fascination for me, a fascination not shared by sensible uncles. It was only much later that I learned to call the stuff Na+ Cl− and that there was an interpenetrating pair of ion lattices underlying their cubical structure which I certainly could not see. Even so it was interesting to stack those grains of salt, the pre-Lego natural material I had to play with. Second answer: I was an undergraduate at Hamilton College. My high school mathematics teacher Florence Deci, who appreciated my art as well as my

Figure 1 The David Borwein Distinguished Career Award of the Canadian Mathematical Society, created in 2004, is a bronze sculpture based on Benson’s formula for the Madelung constant. An exact copy is given to each award winner. Photograph by permission of the sculptor.

Foreword

xiii

maths, had advised me to choose a liberal arts school so I could do both art and science. As a maths undergraduate I was fascinated by summability and conditionally convergent series. From my Hamilton chemistry professor Leland ‘Bud’ Cratty, I first heard about salt and its curious connection with a mathematical sum, Madelung’s constant. The non-commutativity of infinitely many summands was observed by Riemann in relation to the conditionally convergent  series k≥1 (−1)k−1 /k, which is supposed to be a representation of log 2 = 0.69314718 . . .; this value obtains by adding the terms in increasing k order, as is implicit in the convention of summation notation. Even worse in some respects, for Madelung’s series adding terms in increasing cubes gives a different answer than adding terms in increasing balls. So what is the true value, the value with which physicists like Born, Madelung, and Benson and mathematicians like the Borweins would be satisfied? Benson answered this most remarkably with μ = 12π

 2 π m≥0 n≥0 cosh ( 2

1  , (2m + 1)2 + (2n + 1)2 )

which is an absolutely convergent series with all positive terms very rapidly decreasing, affording its evaluation to many decimal places: μ = 1.74756459463318219063621203554439740348516143662474175815282535076504 . . . ,

enough decimals to satisfy this sculptor. Subsequently, Borwein and Crandall [2] and others have learned more and give an almost closed form for μ. When the Borwein family asked me to do a sculpture about summability to celebrate the mathematics of David Borwein and his sons, particularly its application to Madelung’s constant, you can see that my art and science pump had been primed long ago in the deserts of the Rocky Mountains and the forests of the Finger Lakes of upstate New York. It is true that when the Borweins approached me about doing this sculpture, I had been celebrating mathematics with sculpture for decades. However, they approached me while I was in my negative-Gaussian-curvature phase and was carving granite, not salt, and would my geometric negative Gaussian curvature phase be inhospitable to the hard analysis about conditional triple sums over three-dimensional lattices? It happened that I had developed a series of sculptures which involved twodimensional lattice sums, specifically having to do with the planets and Kepler’s third law, i.e., that the squares of the orbital periods of the planets are proportional to the cubes of their radii, when this law is viewed in terms of elliptic complex curves or real tori in four and three real dimensions. For example, the planet Jupiter takes about y = 11 earth–sun years to elliptically orbit the sun at x = 5 earth–sun distances, and 112 = y 2 = x 3 − x + 1 = 53 − 5 + 1 is a perfectly respectable Z-rank-2 elliptic curve in the two complex dimensions of x and

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Foreword

y, which corresponds to four real dimensions. To get the planet Jupiter’s elliptic curve into three real dimensions where I could expect to do sculpture required negative-Gaussian-curvature forms and lattice sums! Some mathematical details behind my negative-Gaussian-curvature phase appeared in ‘Sculpture inspired by work with Alfred Gray: Kepler elliptic curves and minimal surface sculptures of the planets’ [3], reflecting a keynote address by Helaman and Claire Ferguson for the Alfred Gray Memorial Congress on Homogeneous Spaces, Riemannian Geometry, Special Metrics, Symplectic Manifolds and Topology, held in September 2000 in Bilbao, Spain. This work actually made copious use of and reference to the Borwein brothers’ Pi and the AGM [1], an important resource for this negative-curvature phase of my sculpture. What could be more natural than the conditional sum of a three-dimensional lattice as a period of a two-dimensional lattice to create a Madelung triply punctured torus immersed with negative Gaussian curvature in three-dimensional space? The Borwein Award sculpture emerged after considerable computational and sculptural work, which I will sketch next. I had some number-theoretical issues, which I discussed in detail with Jon Borwein. These involved the exponent in the denominator of the lattice sum, s = 12 for the square root. As a function of the complex variable s, ought not the series L NaCl (s) have an analytic continuation to the whole plane, Riemann hypothesis, and even a functional equation? I thought it important to immerse the matter of salt symbolized by Madelung’s constant as L NaCl ( 12 ) in this larger world. Did it have an Euler product? The answers to these two questions are yes, no, yes, and no and appear in the writings of Jon Borwein and others elsewhere. After much computation of L NaCl (s) for various values of s, I settled on μ = L NaCl ( 12 ) and an elliptic curve, y 2 = 4x 3 − (32.6024622677216 . . .)x − (70.6022720835820 . . .) where the decimals correspond to two-dimensional lattice sums for a lattice involving μ, with discriminant −99932.555 . . . The complex variables x, y are complex numbers in four real dimensions and the complex curve equation amounts to two real equations, so that the complex curve is really a surface in four dimensions. There is a dimension embargo (the Planck length is even harder to see than salt lattices!) on sculpture. Sculpture physically resides in spatial three dimensions, hence I enjoy the use of negative curvature to get the geometric surface in four dimensions into the spatial three dimensions where I have much experience. While my aesthetic choice is to carve stone, my award sculptures are in polished silicon bronze. Silicon bronze is an alloy of copper with silicon and a few other things to improve flow and polishing. A typical recipe for silicon bronze is the ‘molecule’ 9438Cu + 430Si + 126Mn + 4Fe + Zn + Pb. I think of the 430Si + 126Mn + 4Fe + Zn + Pb piece as being the ‘stone’ part. I wonder, is

Foreword

xv

there a Madelung constant for this molecule, there being many loose ions in this polycrystalline soup? There are many steps in the casting of silicon bronze, but even before getting to those, I had much computation to do in placing the complex curve into three dimensions as a triply punctured torus. In the course of a computation and developing the computer graphics there are many choices to be made. My decision process is informed by my studio experience in the same way that looking at two-dimensional underwater video material is not at all the same after learning to scuba dive in a three-dimensional environment. This is not the place to discuss all these transitions; there are many. In Figure 2 some of them are shown: computer graphics, wire frame, clay, plaster. There are truly messy in-between parts, especially making of the mould, the wax positive image, the ceramic shells to form a negative flask, and a hot dry throat embedded in sand in which to pour molten bronze; there is the high drama of the pouring of the bronze, the violence of smashing the ceramic flask to release the imprisoned bronze, then the hackingoff of air escape sprues, chasing away all evidence of what violence the bronze has experienced, grinding and sanding the bronze smooth enough to reveal the inevitable natural errors, which must be excavated and welded in kind to prepare for polishing. While the intermediate result is a beautiful polished bronze, shown in Figure 3, this is not the end. I am carving into this silicon bronze the name of each recipient of this elegant CMS–SMC David Borwein Award, the provenance of the sculpture, and also in Figure 1 something about the sculpture relating to salt and summability. This is what is shown for the first recipient. Art is always a social event in the end. In the case of this Borwein Award, the truly priceless part is the awarding of a silicon bronze to celebrate the distinguished careers of gifted people who have given substantial parts of their lives to creating new mathematics and even new mathematicians, as has David Borwein. So far these people have included: 2010: Nassif Ghoussoub 2008: Hermann Brunner 2006: Richard Kane

Figure 2 Stages in the design of ‘Salt’.

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Foreword

Figure 3 The David Borwein Distinguished Career Award of the Canadian Mathematical Society. Photograph by permission of the sculptor.

I am honoured that my mathematical sculpture is part of recognizing and celebrating mathematical lives.

References [1] J. M. Borwein and P. B. Borwein. Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity. Wiley, New York, 1987. [2] J. M. Borwein and R. E. Crandall. Closed forms: what they are and why we care. Not. Amer. Math. Soc., 60(1):60–65, 2013. [3] Helaman Ferguson and Claire Ferguson. Sculpture inspired by work with Alfred Gray: Kepler elliptic curves and minimal surface sculptures of the planets. Contemp. Math., 288:39–53, 2000.

Preface

. . . Born decided to investigate the simple ionic crystal – rock salt (sodium chloride) – using a ring model. He asked Landé to collaborate with him in calculating the forces between the lattice points that would determine the structure and stability of the crystal. Try as they might, the mathematical expression that Born and Landé derived contained a summation of terms that would not converge. Sitting across from Born and watching his frustration, Madelung offered a solution. His interest in the problem stemmed from his own research in Goettingen on lattice energies that, six years earlier, had been a catalyst for Born and von Karman’s article on specific heat. The new mathematical method he provided for convergence allowed Born and Landé to calculate the electrostatic energy between neighboring atoms (a value now known as the Madelung constant).1 Their result for lattice constants of ionic solids made up of light metal halides (such as sodium and potassium chloride), and the compressibility of these crystals agreed with experimental results.2

The study of lattice sums is an important topic in mathematics, physics, and other areas of science. It is not a new field, dating back at least to the work of Appell in 1884, and has attracted contributions from some of the most eminent practitioners of science (Born and Landé [1], Rayleigh, Bethe, Hardy, . . . ). Despite this, it has not been widely recognized as an area with its own important tradition, results, and techniques. This has led to independent discoveries and rediscoveries of important formulae and methods and has impeded progress in some topics owing to the lack of knowledge of key results. In order to solve this problem, Larry Glasser and John Zucker published in 1981 a seminal paper, the first comprehensive review of what was then known about the analytic aspects of lattice sums. This work was immensely valuable to many researchers, including the other authors of the present monograph, but now 1 More exactly, this energy can be obtained from the Madelung constant. 2 From [5], pp. 79–80. Max Born was the maternal grandfather of the singer and actress Olivia

Newton-John. Actually, soon after this they discovered that they had forgotten to divide by 2 in the compressibility analysis. This ultimately led to the abandonment of the Bohr–Sommerfeld planar model of the atom.

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is out of date and lacks the immediate electronic accessibility expected by today’s researchers. Hence, we have the genesis of the present project, the composition of this monograph. It contains a slightly corrected version of the 1981 paper of Glasser and Zucker as well as additions reflecting the progress of the subject since 1981. The authors hope it is sufficiently comprehensive in flavour to be of value to both experienced practitioners and those new to the field. However, as the study of lattice sums has applications in many diverse areas, the authors are well aware that important contributions may have been overlooked. They would thus welcome comments from readers regarding such omissions and hope that internet technology can make this a living and growing project rather than a static compendium. The emphasis of the results collected here is on analytic techniques for evaluating lattice sums and results obtained using them. We will nevertheless touch upon numerical methods for evaluating sums and how these may be used in the spirit of experimental mathematics to discover new formulae for sums. Those interested primarily in numerical evaluation would do well to consult the relatively recent reviews of Moroz [7] and Linton [6]. Several chapters in this monograph are based on published material and, as such, we have tried to retain their original styles. In particular, we have not attempted to iron out the differences in notation (we considered this option but decided it would be very likely to introduce more errors and difficulties than it removed). In particular, we alert the reader that several conventions of the sum mation notation are used liberally throughout the monograph: the symbol may indicate either a single or a multiple sum and the variable(s) and range(s) of summation may be omitted when they are clear from the context. The reader is therefore advised to exercise caution when moving from chapter to chapter and to note that various notations are listed at the beginning of the index. The index uses bold font to indicate entries which are definitions and includes page numbers for the various tables. We have made a full-hearted attempt to correct misprints in the original material. The end-of-chapter commentaries also direct the reader to more recent material and discussions of the source material. In the same spirit, each chapter has its own reference list while a complete bibliography is also provided at the end of the book. Chapter 1 originally appeared as [4]. Chapter 2 originally appeared as [3] and is reprinted with permission from the American Institute of Physics. Chapter 8 originally appeared as [2]: it was published in the Transactions of the American c the American Mathematical SociMathematical Society, in vol. 350 (1998),  ety 1998. Chapter 9 originally appeared as [8]: it was published in the Journal c Springer-Verlag 2011 with kind of Statistical Physics, in vol. 134 (2011),  permission from Springer Science+Business Media.

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Acknowledgments. The authors would like to acknowledge the following: the late Nicolae Nicorovici, who created a TEX version of the original 1981 paper; the Canada Research Chair Programme, the Natural Science and Engineering Research Council of Canada, and the Australian Research Council, which have supported the research of Jon Borwein and Ross McPhedran over many years; also their many colleagues and students who have made important contributions – David McKenzie, Graham Derrick, Graeme Milton, David Dawes, Lindsay Botten, Sai Kong, Chris Poulton, James Yardley, Alexander Movchan, Natasha Movchan, and many others. Chapter 1: The chapter authors wish to thank John Grindlay and Carl Benson for aid with the literature search. Chapter 2: The chapter authors would like to thank Professor O. Knop for originally bringing our attention to the problems involved in lattice sums. John Zucker wishes to thank Sandeep Tyagi for his help and encouragement. Chapter 9: The chapter author wishes to thank Geoff Joyce and Richard Delves, since without their input none of this could have been written. He also thanks Tony Guttmann for many pertinent comments. Commentaries: We would like to thank Tony Guttmann, Roy Hughes, and Mathew Rogers for their valuable feedback on these. Website: The authors are maintaining a website for the book at www.carma. newcastle.edu.au/LatticeSums/, at which updates and corrections can be received.

References [1] M. Born and A. Landé. The absolute calculation of crystal properties with the help of Bohr’s atomic model. Part 2. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, pp. 1048–1068, 1918. [2] D. Borwein, J. M. Borwein, and C. Pinner. Convergence of Madelung-like lattice sums. Trans. Amer. Math. Soc., 350:3131–3167, 1998. [3] D. Borwein, J. M. Borwein, and K. Taylor. Convergence of lattice sums and Madelung’s constant. J. Math. Phys., 26:2999–3009, 1985. [4] M. L. Glasser and I. J. Zucker. Lattice sums. In Theoretical Chemistry, Advances and Perspectives (H. Eyring and D. Henderson, eds.), vol. 5, pp. 67–139, 1980. [5] N. T. Greenspan. The End of the Certain World: The Life and Science of Max Born. Basic Books, 2005. [6] C. M. Linton. Lattice sums for the Helmholtz equation. SIAM Review, 52:630–674, 2010. [7] A. Moroz. On the computation of the free-space doubly-periodic Green’s function of the three-dimensional Helmholtz equation. J. Electromagn. Waves Appl., 16:457–465, 2002. [8] I. J. Zucker. 70 years of the Watson integrals. J. Stat. Phys., 143:591–612, 2011.

1 Lattice sums

Any attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the Spirit of Chemistry. If Mathematical Analysis should ever hold a prominent place in chemistry – an aberration which is happily almost impossible – it would occasion a rapid and widespread degeneration of that science. Auguste Comte Philosophie Positive (1830)

1.1 Introduction It has been more than 100 years since Appell [2] introduced lattice sums into physics, yet the article on which this chapter is based is apparently the first devoted entirely to the subject. We are, of course, aware that parts of other reviews (such as those by Born and Göppert-Mayer [21], Waddington [135], Tosi [133], and Sherman [125]) have dealt with Coulomb sums in ionic crystals, as a casual reading of this chapter will demonstrate. In the perusal of 100 years of literature, we will inevitably have missed or ignored relevant papers and their authors are urged to communicate with us directly. The organization of this review is as follows. In Section 1.2 we present a historical survey, picking out and describing in detail some of the more important methods for calculation. Section 1.3 deals with the representation of lattice sums as Mellin-transformed products of theta functions. In Section 1.4 we discuss the evaluation of two-dimensional lattice sums by number-theoretic means, and in Section 1.5 we examine a promising new application of contour integration. Two brief appendices are concerned with connections among lattice sums, elliptic integrals, and lattice Green’s functions.

2

Lattice sums

1.2 Historical survey Lattice sums are expressions of the form 

F(l)

(1.2.1)

l

where the vector l ranges over a d-dimensional lattice. In this review we shall be concerned primarily with the sums for which F(l) = (exp b · l)|l + g|−s .

(1.2.2)

We shall not indicate the value of d explicitly (although the cases d = 2, 3 will be our main concern) nor shall we dwell on questions of convergence; these details should be clear from the context. Apparently, lattice sums were first considered in physics by Appell [2–4], who examined the form of the solutions to Laplace’s equation for periodic sources. His procedure was to Fourier-transform the potentials and solve the resulting algebraic equations for the coefficients. He obtained an analytical expression for the potential of a line of point charges and derived a variant of the so-called Ewald method for higher-dimensional arrays, but his work appears not to have reached the notice of subsequent workers. In 1902 Epstein [47] reported on efforts to find the most general function satisfying a functional equation similar to that for the Riemann zeta function, ζ (s) =

∞  1 . ns

(1.2.3)

n=1

We describe what we feel are his most important findings. Let q(l) be a positive definite quadratic form in the components of l: q(l) = a11l12 + · · · + add ld2 + 2(a12l1l2 + · · · ), where A = (ai j ), i, j = 1, . . . , d is a symmetric matrix. We shall denote by D the determinant of A and by q(l) the quadratic form defined by the adjoint matrix A = (a i j ),

ai j =

1 ∂D . D ∂ai j

Epstein found that the desired functions were1    g   e−2πih·l    (q; s) = Z  ,  [q(l + g)]s/2  h  l

(1.2.4)

(1.2.5)

where g and h are arbitrary vectors and the sum runs over the d-dimensional integer lattice. The sum is assumed to omit l = −g if g is a lattice vector. This 1 The verticals indicate that the Epstein Zeta function Z depends on the vectors g and h as well as

on the scalars q and s.

1.2 Historical survey

3

class of Epstein zeta functions includes the solutions obtained by Appell and the lattice sums which later became important in crystal physics. Epstein showed that the functions (1.2.5) have the following properties:      g   −g      (i)  (q; s).  (q; s) = Z  (1.2.6) Z      −h   h       g   −g      (ii)  (q; s).  (q; s) = Z  (1.2.7) Z      h   −h  (iii) If any component of h is increased by an integer, the right-hand side of (1.2.5) is unaffected. (iv) If g j is increased by unity, (1.2.5) is multiplied by exp (−2πi h j ). (v) The Epstein zeta functions may be continued analytically beyond their domain of absolute convergence with respect to s (which is Re s > d) by means of the integral representation    g   ∞    −s/2 s/2−1 1   (q; s) = ( 2 s) Z  dz z exp [−π zq(l + g) + 2πil · h] π  1  h  l  ∞ dz z (d−s)/2−1 + D −1/2 exp (−2πig · h) 1  × exp [−π zq(k + h) − 2πig · k]. (1.2.8) k

The sums which occur in (1.2.8) are the generalized theta functions studied extensively by Krazer and Prym [96]. The two sums are related by the analogue of the Jacobi transformation for ordinary theta functions (see below). The lattice of the k vectors is, strictly speaking, the reciprocal lattice for the l vectors, but in the present case these are identical. Equation (1.2.8) is obtained by using the representation  ∞ 1 −s z s−1 e−x z dz (1.2.9) x = (s) 0 and breaking up the range of integration. This simple device, due to Riemann, with minor modifications is the underlying motif in most of the practical work on lattice sums to be discussed below. (vi) The Epstein zeta function obeys the functional equation      g    h   −2πig·h     d −s  (q; d−s).  (q; s) = e √ Z  π −(d−s)/2  π −s/2 ( 12 s)Z    2 D  −g   h  (1.2.10)

4

Lattice sums Equation (1.2.10) is valid for all sets of parameters, whereas (1.2.8) is non-trivial only when g = 0, h = 0. However, if this is not the case, (1.2.8) can be modified and becomes    0   ∞    2 2 −s/2 1   π ( 2 s)Z   (q; s) = dz z (s/2)−1 e−π zq(l) √ − + s (s − d) D 1  0  l  ∞  1 +√ dz z (d−s)/2−1 e−π zq(k) . (1.2.11) D 1 k (vii) When h is not an integer vector, (1.2.5) is an entire function of s; when h is integral, (1.2.5) has a simple pole at s = d:    g    2π d/2  (q; s) = √ + C0 + O(s − d). Z  (1.2.12)  D(d/2)(s − d)  h  Epstein gave expressions for C0 , but these are quite complicated except for the case d = 2, where the result was already known from earlier work by Kronecker and Hurwitz (see Bellman [6]). (viii) The zeta function vanishes for s = −2, −4, −6, . . . and for s = 0, unless g is integral. In this case    g     (q; 0) = − exp (−2πig · h). (1.2.13) Z    h  (ix) If the components of g and h are 0 or 1/2 and 4g · h is odd, then (1.2.5) is identically zero for all s and q.

The numerical evaluation of lattice sums became an important topic in the first two decades of the twentieth century, when X-ray studies, and particularly the work of W. H. and W. L. Bragg, showed that ionic salts consist not of molecules but of interpenetrating lattices of the corresponding ions. The formal aspects and the role of lattice sums in crystal binding and lattice vibration studies were developed extensively by Born [17] and by his students, but the first accurate numerical result pertaining to a three-dimensional crystal – rock salt – was obtained by Madelung [103]. In view of its historical interest, we shall describe his calculation in some detail. Madelung considered the problem of calculating the potential at the origin of an extended array of charges ±E arranged on a rock salt lattice, a portion of which is displayed in Fig. 1.1. The lattice is considered to be composed of vertical planes π0 (through the origin), π±1 , π±2 , . . ., and the plane π0 is considered to be

1.2 Historical survey

5

a a

π2

a

π1

z

x

π0 L−1

y

O L0

(a)

L1

+E −E (0, 1, 1)

(1, 1, 1)

(0, 0, 1) (1, 0, 1) a/2

(1, 1, 0) (0, 1, 0) (b) (0, 0, 0)

(1, 0, 0)

Figure 1.1 (a) The rock salt structure decomposed into planes and lines of charges. (b) Unit cell of the rock salt lattice.

formed from vertical lines L 0 (through the origin), L ±1 , . . . Hence, by symmetry, the potential at O is V (O) = VL 0 (O) + 2

∞ 

VL n (O) + Vπn (O) .

(1.2.14)

n=1

Thus the problem reduces to calculating the potential due to a line of alternating charges and a centred square planar array of alternating charges. (The prime indicates that the charge E is absent from the origin on the line L 0 .) Although unaware, apparently, of the earlier work by Appell, Madelung used precisely the same method. Consider first the potential due to the array of charges shown in Fig. 1.2a. An arbitrary linear periodic charge density can be written ρ(x) =

∞  −∞

ρl e2πilx/a .

(1.2.15)

6

Lattice sums a E

E

−a

O

E

E

a

2a

x

r (x, r) (a) a −E

+E

−E

+E

−E

+E

−E x

O r (x, r) (b)

Figure 1.2 (a) Linear array of constant charges. (b) Alternating linear array.

Accordingly, we look for the potential at (x, r ) in the form v1 (x, r ) =

∞ 

Fl (r )e2πilx/a .

(1.2.16)

−∞

By inserting (1.2.16) into Laplace’s equation, we find d 2 Fl 4πl 2 1 d Fl − + Fl (r ) = 0. r dr dr 2 a2

(1.2.17)

The solution to (1.2.17) which vanishes as r → ∞ (l = 0) is  Fl (r ) =

Cl K 0 (2πlr/a), l = 0, l = 0, C0 ln (2a/r ),

(1.2.18)

where K 0 denotes the modified Bessel function. However, from Poisson’s equation we have  1 ∂v1  , (1.2.19) ρ(x) = − 2 ∂r r =0 which gives Cl = 2ρl . Hence the desired potential is  v1 (x, r ) = 2ρ0 ln

2a r

 +2

∞  −∞

 ρl K 0

2πlr a

 e2πilx/a .

(1.2.20a)

(Note that we have adjusted the potential by an additive constant such that as r → ∞ it gives the potential of a uniform line charge of density ρ0 .) For the

1.2 Historical survey

7

linear array in Fig. 1.2a Madelung’s result is ∞ 

E 2 + r 2 ]1/2 [(x − a) −∞  

  ∞  2πlr 2πlx 2E 2a +2 cos = ρl K 0 ln . (1.2.20b) a r a a

v1 (x, r ) ≡

l=1

Mathematically, this procedure is equivalent to using the Poisson summation formula, by which the slowly convergent first series in (1.2.20b) is replaced by the rapidly convergent second series. The potential for the alternating array in Fig. 1.2b is v L (x, r ) = v1 (x, r ) − v1 (x + a/2, r )     ∞ 2π(2l − 1)r 2π(2l − 1)x 8E  = K0 cos . a a a

(1.2.21)

l=1

The potential at O due to all the other charges is most easily obtained by direct summation:   1 1 1 4E 4E  1 − + − + ··· = ln 2. (1.2.22) v L (O) = a 2 3 4 a In the case of the plane rectangular lattice shown in Fig. 1.3 we begin by looking for a potential having the form  2πl2 x 2πl1 x cos (l = (l1 , l2 )). V = Fl cos (1.2.23) a b l

Substituting this into Laplace’s equation yields d 2 Fl − 4π 2 kl2 Fl = 0, dz 2

kl2 =

l12 l22 + ,n a2 b2

b

z y

O

x a

Figure 1.3 Planar rectangular grid of point charges.

(1.2.24)

8

Lattice sums

which has the solution (vanishing as z → ∞ for kl2 = 0)  C1 e−2π zkl , kl = 0, Fl = C0 + C1 z, kl = 0. This corresponds to the periodic charge density  2πl2 x 2πl1 x ρ(x, y) = cos . ρl cos a b

(1.2.25)

(1.2.26)

l

Proceeding from Poisson’s equation, just as for the linear case, we find v2 (x, y, z) =

∞ 4E  e−2π kl z 2πl1 x 2πl2 y cos cos ab kl a b l1 ,l2 =1   ∞ 2E  1 −2πlz/a 2πlx 2πly −2πlz/b +e + cos cos e b l a b l1

2π E z + const. − ab

(1.2.27)

For a centred square alternating array, corresponding to the planes π in Fig. 1.1, we obtain easily Vπ =

∞ ∞ 16E   e−2π zkl 2πl1 x 2πl2 y cos cos . ab kl a a

(1.2.28)

l1 =1 l2 =1

In addition to those above, Madelung’s paper contains a number of other calculations relating to two-and three-dimensional periodic arrays. Using formulas (1.2.21), (1.2.22), and (1.2.28), it is simple to calculate the potential at the reference site O for the rock salt structure. On the basis of (1.2.14), for three-decimal place accuracy Madelung found it sufficient to keep the following contributions to v(O) = (E/a) μ. The line L 0 , μ0 = 4 ln 2. The lines L ±1 (x = 0, r = a/2), μ1 ∼ = 16K 0 (π ) + K 0 (3π ); The lines L ±2 (x = 0, r = a), μ2 ∼ = −16K 0 (2π ). The lines L ±3 (x = 0, r = 3a/2), μ3 ∼ = 16K 0 (3π ). The planes π±1 (x, y, z) = (0, 0, ±a/2), √ √ exp (−π exp (−π 2) 10) +2 μ4 ∼ . √ √ = 32 10 2 √ exp (−π 2) ∼ . (6) The planes π±2 (x, y, z) = (0, 0, ±a), μ5 = −32 √ 2

(1) (2) (3) (4) (5)

Thus Madelung obtained μ ∼ = 3.487.

1.2 Historical survey

9

Although very expedient in this case, Madelung’s procedure is not generally applicable, owing to the complicated geometrical considerations needed to decompose more complex lattice structures. Indeed, Landé [98] overlooked a line of charges in decomposing the fluorite lattice. Madelung’s method was later applied by Bethe [14], Jones and Dent [84], and others to obtain field distributions in NaCl and other ionic crystals. It is interesting that, by summing directly over the first nine shells of ions surrounding a sodium ion, Kendall [94] was able to obtain Madelung’s result to within 10%. In this review we shall not consider such direct summation procedures. Curiously, almost exactly when Madelung was doing his work, Ornstein and Zernicke [112] independently used precisely the same method in three dimensions to re-sum similar series occurring in a study of the magnetic properties of cubic lattices. The need for a more flexible summation procedure applicable to an arbitrary crystal structure was met by P. Ewald [49]. The procedure is based on the rediscovery of the Riemann–Appell device expressed in Equation (1.2.11) and was formulated and developed for cubic crystals in terms of ordinary theta functions in Ewald’s 1912 thesis. He later published a generalized version, which, because of its usefulness and wide application, we shall describe in detail. (For an exposition relating to d = 2, see Fetter [50].) Consider a crystal lattice described by the vectors R p = Rl + rt

p = (l, t),

t = 1, . . . , ν

(d = 3).

(1.2.29)

Here l is an integer vector which labels the Bravais lattice vectors Rl ; the vectors rt lie in the unit cell and are associated with electric charges εt . We shall denote by Kl the reciprocal lattice vectors corresponding to the Rl ; i.e., if Rl = l1 a1 + l2 a2 + l3 a3 then Kl = l1 b1 + l2 b2 + l3 b3 , where ai · b j = δi j ,

V0 = a1 × a2 · a3 .

We shall illustrate the procedure for the electrostatic sum   ε p   exp (i K 0 R pp ) exp (ik · R p ),  ( p) = R pp 

(1.2.30)

p = p

where R pp = |R p − R p |. We have   ν   εt  exp (ik · rt  ) [π(r − rt  ) − δtt  Fl (r − rt  )] , (1.2.31)  ( p) = lim r→R p

t  =1

10

Lattice sums

where π(r) =

 exp (i K 0 |Rl − r| + ik · Rl ) |Rl − r| 

(1.2.32)

exp (i K 0 |Rl − r| + ik · Rl ) . |Rl − r|

(1.2.33)

l

and Fl (r) =

Hence, the problem reduces to calculating the slowly convergent sum (1.2.32) and taking the limit in (1.2.31). Precisely as Epstein [47] did, Ewald used the following basic theorem from Krazer and Prym [96]: Theorem 1.2.1 Let d1 , d2 , d3 generate a Bravais lattice, let A and v be arbitrary vectors and let V0 = d1 × d2 · d3 . Let c1 , c2 , c3 be the corresponding unit vectors for the reciprocal lattice and let   ql = li di + v, pl = π li ci . (1.2.34) Then 

exp (−ql2 + iql · A) =

l

π 3/2  exp [−(pl − V0

1 2

v)2 + 2ipl · A],

(1.2.35)

l

where the sum is over the integer l lattice. The sum in (1.2.32) is only conditionally convergent. To avoid technical difficulties attendant on this fact, it is advisable to render the series absolutely convergent by including a convergence factor. This is done most simply by assuming that K 0 has an imaginary part that is small and positive. To transform (1.2.32) we take note of the integral representation (1.2.10), for α ≥ 0,  ∞ 2 e−α R 2 2 2 2 =√ e−R t e−α /4t dt. (1.2.36) R π 0 We can replace α by −i K 0 if the path of integration is deformed in such a way that K 02 /4t 2 becomes purely imaginary as t approaches zero. Thus we have    ∞ K 02 exp (i K 0 R) 2 2 2 exp −R t + 2 dt, (1.2.37) =√ R 4t π (0) where the path of integration leaves the origin along the ray arg t = arg K 0 − π/4 and then returns to the real axis. From (1.2.32) and (1.2.37) we have

 ∞ K 02 2 2 2 exp −(Rl − r) t + ik · Rl + 2 dt. (1.2.38) π(r) = √ 4t π (0) l

1.2 Historical survey

11

Next, by (1.2.35) with ci = (t/π )ai , di = (π/t)bi , v = 2tr, so that ql = we also have 2π π(r) = V0



∞ (0)

l

1 kl (2π Kl + k) ≡ ; 2t 2t

−(k2l − K 02 ) dt exp + ikl · r 3 . 2 4t t

(1.2.39)

The sum in (1.2.38) converges rapidly when t is large, while that in (1.2.39) converges best when t is small. Instead of dividing the range of the t-integration at 1, as Epstein did, Ewald introduces the adjustable parameter η (assumed real) and writes π(r) = π (1) (r) + π (2) (r),

 −(k2l − K 02 ) dt 2π η  (1) exp + ikl · r 3 , (1.2.40) π (r) = 2 V0 (0) 4t t l

 ∞ K 02 2 (2) 2 2 π (r) = √ exp −(Rl − r) t + ik · Rl + 2 dt. 4t π η l

The path of integration in π (1) can be returned to the real axis and the integral is elementary, while the integral in π (2) leads to the error function. In this way, we obtain 4π  exp [−(k2l − K 02 )/4η2 + ikl · r] , (1.2.41) V0 k2l − K 02 l     exp (ik · Rl ) i K0 Re exp (i K 0 |Rl − r|) erfc |Rl − r|η + . π (2) (r) = |Rl − r| 2η π (1) (r) =

l

We also have, from (1.2.33) and the fact that erfc(x) = 1 − erf(x), for the term t = t  (rt = 0) in (1.2.31) π(r) − Fl (r) = π (1) (r) + π (2) (r), sin K 0 |Rl − r| π (1) (r) = π (1) (r) − |Rl − r|    1 i K0 Re exp (i K 0 |Rl − r|) erf |Rl − r|η + , − |Rl − r| 2η π (2) (r) = π (2) (r)    i K0 exp (ik · Rl ) Re exp (i K 0 |Rl − r|) erfc |Rl − r|η + . − |Rl − r| 2η (1.2.42)

12

Lattice sums

There is now no difficulty in taking the limit in (1.2.31). The quantity η is to be chosen so that both series converge rapidly. For the case of a Madelung sum we take both K 0 and k → 0, so  ( p) → V (R p ) =

ν  t  =1

π (1) =

2εt η εt  (π (1) + π (2) ) − √ , π

1  exp [−π 2 K l2 /η2 + 2πiKl · (rt − rt  )] , π V0 K l2

(1.2.43)

l

π (2) =

 erfc(|R p − R p |η) . |R p − R p |  p

The prime on the sum π (1) indicates that the l = (0, 0, 0) term is omitted because, for an ionic crystal, ν  εt  = 0. t  =1

We have also used the fact that exp (2πiKl · R p ) = 1. Equations (1.2.43) are very useful for practical calculations, as we shall see by evaluating the Madelung constant for NaCl. We have V (O) = V (1) (O) + V (2) (O), 1  exp (−π 2 K l2 /η2 ) 2ηE V (1) (O) = Sl + √ , π V0 π K l2 l=0

Sl =

2 

εt exp (−2πiKl · rt ), n

t=1

V (2) (O) =

2  

εt

t=1 l=0

erfc(|Rl + rt |η) . |Rl + rt |

For the NaCl structure (see Fig. 1.1) a1 =

1 2

b1 =

1 a (1, 1, 1),

a(0, 1, 1),

a2 =

1 2

b2 =

1 a (1, 1, 1),

V0 =

1 4

a(1, 1, 0),

a3 =

1 2

b3 =

1 a (1, 1, 1).

a(1, 0, 1),

a3

Hence, 1 a(l2 + l3 , l1 + l2 , l1 + l3 ), 2 1 Kl = (−l1 + l2 + l3 , l1 + l2 − l3 , l1 − l2 + l3 ). a Rl =

(1.2.44)

1.2 Historical survey

13

The basis in the unit cell is r1 = (0, 0, 0)

(+E),

r2 =

1 a(1, 1, 1) 2

(−E).

It is convenient to replace η by η/a. To evaluate V (1) (O), we note that Sl =

 2E, l1 + l2 + l3 odd, 0

otherwise.

Thus we sum only over those reciprocal lattice vectors for which l1 +l2 +l3 is odd. For example, the six vectors l = [1, 0, 0] together with (1, 1, 1) and (−1, −1, −1) give the eight reciprocal lattice vectors aKl = [1, 1, 1] for which a 2 K2l = 3. In this way, we easily compile the following table: a 2 K2l

Number

3 11 19

√ (η = 2/ π )

Summand in V (1) (O)

8 24 24

0.3218078 0.0004917 0.0000005

Hence V (1) (O) ∼ = (−E/a)(3.3554000). By inspection it can be seen that the term t = 1 in V (1) (O) involves a sum over the vectors m = (l2 + l3 , l1 + l2 , l1 + l3 ) for which m 1 + m 2 + m 3 is even, while the term t = 2 correspondingly involves the vector m = (l2 + l3 + 1, l1 + l2 + 1, l1 + l3 + 1) for which the sum of the √ components is odd. Therefore we have (for η = 2/ π) V

(2)

√ 2E  m 1 +m 2 +m 3 erfc( π m) . (O) = (−1) a m

(1.2.45)

m=0

We easily construct the corresponding table: m [1, 0, 0] [1, 1, 0] [1, 1, 1] [2, 0, 0]

Number 6 12 8 6

m

Sign

√1 √2 3 2

− + − +

Summand in V (2) (O)

√ (η = 2/ π )

−0.1462648 +0.0066688 −0.0001320 +0.0000034

from which we have V (2) (O) = −(E/a)(0.13972471) and μ = 3.49512. The √ value η = 2/ π has been chosen so both series converge at roughly the same rate. Although it is not of direct concern, it is interesting to consider a paper by Born [18], where the following problem is discussed: Arrange equal numbers of positive and negative charges on a simple cubic lattice of finite extent so

14

Lattice sums

the electrostatic energy is a minimum. Born treats this problem subject to the following simplifications: (1) periodic boundary conditions are used; (2) the Coulomb potential is replaced by e−κr /r with κ → 0 eventually; (3) Ewald’s method is used for computation. Again, we are dealing with an integer lattice, but the charges at the lattice sites have period N in all directions: εl+(N ,0,0) = εl+(0,N ,0) = εl+(0,0,N ) = εl ;

(1.2.46)

we have denoted the sites in the ‘cell’ by l0 , . . . , l N −1 . By electrical neutrality N −1 

εl j = 0.

j=0

The electrostatic potential at l j is j =

 εl+l j l=0

l

e−κl ,

(1.2.47)

so the electrostatic energy per cell becomes =

N −1 1 1  e−κl εl j  j = sl , 2 2 l

(1.2.48a)

l=0

j=0

sl =

N −1 

εl j εl+l j .

(1.2.48b)

j=0

Note that  is equivalent to the potential at the origin of an infinite lattice with ‘charges’ sl at each site. These charges have the properties sl+(N ,0,0) = sl+(0,N ,0) = sl+(0,0,N ) = sl = s−l . Consequently they have the Fourier representation sl =

N −1 

ξ j cos

j=0

2π (l · l j ), N

(1.2.49)

where ξj =

N −1 1  2π (l j · li ). sli cos 3 N N i=0

(1.2.50)

1.2 Historical survey

15

N −1 1  ξj j, N3

(1.2.51)

Therefore (letting κ → 0) =

j=0

where j =

1 3  cos (2π/N )(l · l j ) N , 2 l

j = 0, 1, . . . , N − 1,

(1.2.52)

l=0

are called normal potentials. It was suggested by Born, and later made explicit by Emersleben [40] and Hund [81], that any ionic crystal can be approximated by a simple cubic lattice with point charges at various ‘rational’ sites in the unit cell. Hence all Madelung constants can be evaluated simply as finite sums once one knows the value of the function (Born’s Grundpotential) 1 cos 2π(l · z). (1.2.53) π(z) = l l=0

This has the periodicity properties π(z 1 + 1, z 2 , z 3 ) = π(z 1 , z 2 + 1, z 3 ) = π(z 1 , z 2 , z 3 + 1),

(1.2.54)

π(z) = π(−z), and so by symmetry need only be calculated in one-fortyeighth of the cubic unit cell. On the basis of these considerations, Born showed that the NaCl structure represents a relative minimum for the electrostatic energy. There is yet no general proof that it provides the absolute minimum. (Born did prove this for one dimension.) Born turned over the problem of evaluating the Grundpotential to his student Emersleben, who devoted his thesis to this topic, and the results were summarized in two papers in 1923 [40, 41]. In the first paper Emersleben extended the Grundpotential idea to an arbitrary Bravais lattice and to the case of a general inverse power law; he showed these quantities to be simple Epstein zeta functions:    0    π(z) = Z   (q; s),  z  where the form q is given by the unit vectors of the lattice, q = (ai j ) = (ai · a j ). In addition, Emersleben [40] generalized Epstein’s integral representation (1.2.8) slightly, by using Ewald’s idea of decomposing the range of the z-integral at z = η rather than z = 1. It is interesting that this is the first publication in which Epstein’s work is referred to in the context of ionic lattice sums, although apparently no

16

Lattice sums

use was made of his results other than to assess continuity with respect to s. In his second paper, Emersleben [41] applied Ewald’s procedure to calculate π(m 1 /12, m 2 /12, m 3 /12), 0 ≤ m i ≤ 6, and used the results to find the Madelung constants for several cubic crystals. For example, for the NaCl structure shown in Fig. 1.1b, we have eight ions in a unit cell with +E:

l0 l1 l2 l3

= (0, 0, 0) = (0, 1, 1) = (1, 0, 1) = (1, 1, 0)

−E:

l4 l5 l6 l7

= (1, 1, 1) = (1, 0, 0) = (0, 1, 0) = (0, 0, 1)

ξ0 = ξ1 = ξ2 = ξ3 = ξ= ξ6 = ξ7 = 0,

ξ4 = 8E.

From (1.2.51) and (1.2.53), 1  ξj π =− 4 E 7

μNaCl

j=0



l j1 l j2 l j3 , , 2 2 2



 = −2π

1 1 1 , , 2 2 2



= 3.495115. In this way Emersleben [41] found the following values: Crystal

Madelung constant μ

ZnS

7.56584 = π( 14 , 14 , 14 ) + 34 π(0, 0, 12 ) + 14 π( 12 , 12 , 12 )

CaF2

5.81828 = 12 π( 14 , 14 , 14 ) + 34 π(0, 0, 12 ) + 14 π( 12 , 12 , 12 )

CsCl

2.03536 = 32 π(0, 0, 12 ) + 12 π( 12 , 12 , 12 )

Cu2 O

3 π(0, 0, 1 ) + 3 π(0, 0, 1 ) + 1 π( 1 , 1 , 1 ) 4.75219 = 32 4 2 2 4 2 2 2 3 π(0, 1 , 1 ) + 1 π( 1 , 1 , 1 ) + 34 π(0, 14 , 14 ) + 16 2 4 8 4 4 4 3 1 1 1 3 π( 1 , 1 , 1 ) + 4 π( 2 , 2 , 2 ) + 32 4 2 2

As pointed out by Hund [81] the result for Cu2 O is incorrect, but it has often found its way into the modern literature. The correct result is given in Table 1.2 below. At the same time another of Born’s students, H. Kornfeld [95], adapted Ewald’s procedure to the calculation of the electrostatic energy of multipole lattices. The Ewald procedure continues to be the most widely used numerical scheme for calculating lattice sums and particularly Madelung constants. A survey of numerical values obtained in this way before 1932 was given by Sherman [125] (see also Waddington [135] and Tosi [133]). A third approach to calculating lattice sums was given by Jones [83] and Jones and Ingham [85], who were concerned with the evaluation of Epstein zeta

1.2 Historical survey

17

functions for large integral s, especially for the cubic lattices. These arise naturally in examing the stability of rare gas crystals whose atoms interact via the 6–12 potential, for example.2 The method is based on E. Landau’s results [97] relating to the distribution of lattice points within ellipsoids and particularly on the formula  {N − q(l + q)}σ e2πih·l ν(N − q) (1.2.55) Q σ (N ) ≡ l

=

(σ + 1){h} √ π d/2 N σ +d/2 − {g} N σ e2πig·h + Q σ (N ), 1 (σ + 1 + 2 d) D

where Q σ (N ) = e−2πig·h

N (σ/2)+(d/4)  e−2πig·k Jσ +d/2 [2πq(k + h)N ]1/2 √ [q(k + h)]σ/2+d/4 πσ D k (1.2.56)

and Jν denotes the Bessel function of the first kind. Here ν(N − q) denotes that the sum is restricted to vectors l for which q(l + g) is less than N ; {h} is 1 if h is an integer vector and vanishes otherwise. From the fact that  N  s  2πih·l e2πih·l ν(N − q)(q −s/2 − N −s/2 ) = e ν(N − q) u −s/2−1 du 2 q l l  N s u −s/2−1 Q 0 (u) du, (1.2.57) = 2 0 we immediately have the representation    q   ∞    (q; s) = s Z  u −s/2−1 Q 0 (u) du  2 0  h  and 

e

2πih·l

ν(N − q) q

−s/2

l

=N

−s/2

s Q 0 (N ) + 2



N

(1.2.58a)

u −s/2−1 Q 0 (u) du.

0

(1.2.58b) By subtracting the last two relations we obtain    g   ∞    (q; s) = Z N − N −s/2 Q 0 (N ) + s u −s/2−1 Q 0 (u) du, Z   2 0  h   e2πih·l ν(N − q) q −s/2 . (1.2.59) ZN = l 2 The 6–12 potential is given by V (r ) = A/r 6 + B/r 12 , where A and B are material constants. 12 12 12

18

Lattice sums

Next we integrate by parts p times, which gives   g  p     n + s/2 − 1   Z   (q; s) = Z N − N−s/2−n Q n (N ) n h n=0  s/2 + p − 1  ∞ s +p u − p−s/2−1 Q p (u) du. (1.2.60) + 2 p 0 Now using Landau’s representation (1.2.55) we obtain Jones’ and Ingham’s working formula    g      p + s/2 − 1 {h} p!π d/2 (s + 2 p)   N −(s−d)/2 Z  (q; s) = Z N + ( p + d/2 + 1)√ D(s − d) p  h     p + s/2 − 1 −2πig·h s + 2 N −s/2 − {g}e s p  p   n + s/2 − 1 Q n (N )N −s/2−n + R p (N , s), (1.2.61) − n n=0

where R p is due to the term Q p (N ) in (1.2.56). Finally, technical lemmas are provided which in essence show that, for p > d/2, N > 12, p + d/2 − 1/2 < 6,    h     s(s + 2) · · · (s + 2 p)  q; p + d + 3 . Z  |R p (N , s)| < √ 2 2 2 p π p+2 D N s/2+ p/2−d/4+3/4  0  (1.2.62) The corrections to the finite sum Z N on the right-hand side of (1.2.61) are used merely to estimate the value of N for which Z N gives the zeta function to the required accuracy. For s ≥ 6, N = 25 was found to be sufficient for six-place accuracy. To evaluate Z N and Q σ (N ), number-theoretical tables were used for the quantity an (q) = number of representatives of n by the quadratic form q, so that ZN =

N 

n −s/2 an (q),

n=1

(1.2.63) N  Q σ (N ) = (N − n)σ an (q). n=1

In all the cases considered these series required that no more than five terms be retained for the necessary accuracy.

1.2 Historical survey

19

As an example of the use of these formulae we shall work out the sum  R −12 S= for a face-centred cubic lattice, in which case  S= [(l1 + l2 )2 + (l2 + l3 )2 + (l1 + l3 )2 ]−6 l=0

 =

26 a 12



   0    Z   (q; 12),  0 

(1.2.64)

where q(p) = l12 +l22 +l32 +l1l2 +l2l3 +l3l1 . (Thus q(l) = 32 l12 + 32 l22l32 −l3l1 −l1l2 , D = 12 .) Using p = 3 (we estimate z| 00 |(q; 6) < 20) we easily find that |R3 | < 106 with N = 10. Hence, to calculate Z 10 and Q n (10), we require the first ten values of an (q). But it is clear that q(l) = 12 (m 21 + m 22 + m 23 ), where m 1 = l1 + l2 , m 2 = l2 + l3 , m 3 = l3 + l1 so that the sum m l + m 2 + m 3 is even. Therefore an (q) is the number of ways in which 2n can be expressed as the sum of three squares (since m 1 + m 2 + m 3 is even only if m 21 + m 22 + m 23 is even, this restriction can be dropped). By inspection we have a1 a2 a3 a4 a5

= 12, = 6, = 24, = 12, = 12,

a6 a7 a8 a9 a10

= 8, = 48, = 6, = 36, = 48.

Thus from (1.2.63) Z 10 = 12.13109, Q 0 (10) = 164, and therefore

Q l (l O) = 680,

Q 2 (10) = 3884,

Q 3 (10) = 26036

   0     (q; 12) = 12.13137. Z    0 

This method is reasonable for simple lattice structures if a desk calculator is used, but does not appear to be suitable for large-scale computation. Extensive use of Jones’ and Ingham’s results was made by Hund [80] in his study of the stability of cubic lattices. Hund appears to have been the first to use the expedient of estimating the remainder, after summing over the first few shells of nearest neighbours, by replacing the sum by an integral. This procedure was developed more carefully by Evjen [48], but its accuracy is questionable. Finally, Topping [132] modified Jones’ and Ingham’s procedure to evaluate several two-dimensional lattice sums. However, as will be discussed later in detail, all such sums can be

20

Lattice sums

evaluated exactly. Indeed, precisely the sums considered by Topping had been exactly evaluated by Lorenz [100] more than 50 years earlier. The lattice summation problem does not appear to have attracted much attention over the decade from 1930 to 1940. In spite of excellent reviews of the subject by Born and Göppert-Mayer [21] and by Højendahl [77], and extensions of the Ewald procedure such as that by Fuchs [51], who used it in calculating the electrostatic energy in metallic copper, as late as 1939 Orr wrote: ‘Since no simple analytic method is available, for evaluating these sums, they were summed numerically over at least the nearest 250 points and the remaining contribution determined by integration.’ The most significant work during this period was Hund’s [81] extension of Born’s Grundpotential concept. Hund’s basic idea was to decompose a given lattice structure into ν Bravais lattices, each of which consists of identical ions. He then calculated the potential at a given site due to each of these simple lattices and summed over these contributions to obtain the final result. However, since the component sums are individually divergent, the corresponding lattice was assumed to be rendered electrically neutral by a uniform charge distribution of the appropriate sign. These fictitious charge distributions naturally cancel and make no contribution to the final result. This procedure has the advantage that, since for a given lattice system the lattice constants can be scaled out of the individual sums, the lattice sums for all crystals can be expressed in terms of a few basic dimensionless ones. Thus, relations between the sums for the three basic cubic lattices, for example, can be discerned immediately. Secondly, once these basic sums have been calculated, e.g., by the Ewald method, the lattice sums for any cubic crystal can be evaluated. This is also the case, with minor modification, for other crystal systems. For example, in the case of hexagonal crystals the basic sums depend on the c/a ratio, which must be the same for crystals whose lattice sums are to be compared. Furthermore, symmetry can be exploited to reduce the number of sums to be calculated even further. To illustrate this, consider the Madelung sum S = t

ν   l

t=1

eZ t , |Rl + rt − rt  |

(1.2.65)

where the prime indicates that the term corresponding to the vanishing of the denominator is omitted and e denotes the electronic charge. This sum can be expressed in the form 

e Z t  ψ(0, 0, 0) + St  = Z t ψ(xt , yt , z t ) , (1.2.66) a  t=t

where, in terms of the unit vectors, a1 + yt a2 + z t a3 , rt − rt  = xt

1.2 Historical survey

21

is a point in the unit cell. Here ψ(x, y, z) is the potential at the point a −1 (x a1 + a3 ) of a Bravais lattice of unit charges at y a2 + z 1 a1 + l2 a2 + l3 a3 ) (l1 a neutralized by a uniform negative charge distribution. The potential ψ(0, 0, 0) is the same quantity but with the term l = (0, 0, 0) omitted. By the use of symmetry it is easily shown that ψ(x, y, z) (see, e.g., Tosi [133]) need be calculated only in some irreducible symmetry element of the unit cell. The quantity ψ(0, 0, 0) can be eliminated for the cubic lattices as follows. The simple cubic lattice of side a can be looked on as composed of eight simple cubic lattices of side 2a. The coordinates of a point (x, y, z) in the unit cell of the former has the respective coordinates in the unit cell of the latter ξ1 ξ2 ξ3 ξ4

= ( 12 x, 12 y, 12 z), = ( 12 (x − 1), 12 y, 12 z), = ( 12 x, 12 (y − 1), 12 z), = ( 12 x, 12 y, 12 (z − 1)),

ξ5 ξ6 ξ7 ξ8

= ( 12 (x − 1), 12 (y − 1), 12 z), = ( 12 (x − 1), 12 y, 12 (z − 1)), = ( 12 x, 12 (y − 1), 12 (z − 1)), = ( 12 (x − 1), 12 (y − 1), 12 (z − 1)). (1.2.67)

Therefore we have Hund’s identity 1 ψ(ξk ). 2 8

ψ(x, y, z) =

(1.2.68)

k=1

In particular, ψ(0, 0, 0) = 12 [ψ(0, 0, 0) + ψ( 12 , 0, 0) + ψ(0, 12 , 0) + ψ(0, 0, 12 ) + ψ( 12 , 12 , 0) + ψ( 12 , 0, 12 ) + ψ(0, 12 , 12 ) + ψ( 12 , 12 , 12 )]. (1.2.69) Hence, by cubic symmetry, ψ(0, 0, 0) = 3ψ( 12 , 0, 0) + 3ψ( 12 , 12 , 0) + ψ( 12 , 12 , 12 ). This procedure has been significantly elaborated and extended and will be discussed in more detail in Section 1.3. In the early 1940s a number of lattice sums were calculated by Born and coworkers (Born and Fürth [20]; Born and Misra [22]), by Misra [106], and by Peng and Powers [113], principally by the Ewald technique. One exception is the work of Born and Bradburn [19], who calculated a variety of sums having the form Snk (α) =

 l k1 l k2 l k3 1 2 3 n

l=0

l

exp (−il · α)

(1.2.70)

for the face-centred cubic lattice. These were obtained by means of the identity Snk (α) = i k1 +k2 +k3

∂ k1 +k2 +k3 ∂αk11 ∂αk22 ∂αk33

Sn0 (α),

(1.2.71)

22

Lattice sums

so that only Sn0 (α) need be evaluated in detail. The latter was treated by a variant of the Epstein–Ewald method, as we now illustrate for the simple cubic lattice. By using the representation (1.2.9) we have  ∞  1 2 Sn0 (α) = e−il·α e−l μ μn/2−1 dμ (n/2) 0 l=0  ∞ n/2  (π ) 2 β n/2−1 e−πβl −il·α dβ. (1.2.72) = (n/2) 0 l=0

In terms of Jacobian theta functions, the sum above is 3  i=1

θ3

α

i

2

 , e−πβ − 1 = σ (β).

(1.2.73)

Next, we break up the integral into a part from 0 to 1 and a part from 1 to ∞. The part from 1 to ∞ may be obtained directly from the integral in (1.2.72), but for the part from 0 to 1 we first use Jacobi’s transformation θ3 (α, e−πβ ) = β −1/2 exp (−α 2 /πβ) θ3 (iα/β, e−πβ ).

(1.2.74)

We now have   π n/2 2 S0 + S1 − , (n/2) n  ∞ S0 = β n/2−1 σ (β) dβ,

Sn0 (α) =



1 1

S1 = 0

(1.2.75)

   3  iα j −πβ α2  . ,e β n/2−5/2 exp − θ3 4πβ 2β j=1

The integral S0 is given by S0 =

∗



πl 2 cos (l · α) 4

−n/2

 

n πl 2 , 2 4

 ,

(1.2.76)

where the symbol ∗ denotes that only one of l and −l is included and (a, x) is the incomplete gamma function. Next we have, with β −1 → t,  2   ∞ t α dt + lπ t −n/2+1/2 exp − S1 = π 2 1 l   [(α/2) + lπ ]2 (n−1)/2  1 n (α/2 + lπ )2  = . (1.2.77)  − , π 2 2 π l

1.2 Historical survey Thus Sn0 (α)

23

  2n+1 ∗ n πl 2 −n , = cos (l · α) l  (n/2) 2 4   1 n (α + 2lπ )2 (2π )1/2  n−1 (α + 2lπ )  + n − , 2 (n/2) 2 2 4π l

2π n/2 − . n(n/2)

(1.2.78)

The derivatives of this expression with respect to the components of α are easily worked out with the aid of a recursion relation for the incomplete gamma function: ∂ (a, x) = a(a, x) − (a + 1, x). (1.2.79) ∂x The resulting sums converge reasonably rapidly. This procedure was adapted by Heller and Marcus [75] to treat a class of dipole sums. In 1952 Bertaut [11] derived and generalized Ewald’s summation formula by the following procedure, which has not yet been fully exploited. We are interested in calculating the electrostatic energy of a lattice of point charges q j located at r j = Rl + xk , where the Rl form a Bravais lattice and the xk lie in the unit cell. Denote the reciprocal lattice vectors by h. Then the electrostatic energy is x

W = Wi + Ws ,  qi q j 1 Wi = , ri j = |ri − r j |, 2 ri j

(1.2.80)

i= j

and Ws is the (infinite) self-energy. By a well-known theorem in electrostatics, W is unchanged if the point charges are replaced by non-overlapping spherical charge densities:  σ (r ) dr = 1. (1.2.81) q j δ(r ) → q j σ (r ), Then the charge density is ρ(x) =



q j σ (x − r j ).

(1.2.82)

j

Let φ(q) =

 F(q) =

σ (x) e2πiq·x dx,



qk e2πiq·xk

(the structure factor).

k ∈ cell

Then



ρ(x) ρ(x + y) dx dy x 1  |F(h)|2 = |φ(h)|2 , 2π V h2

1 W = 2

(1.2.83)

24

Lattice sums

where V is the volume of the unit cell. Alternatively,  σ (x − r j ) σ (x + y − ri ) 1 W = dx dy qi q j 2 |x − r j | i, j   1 du = qi q j dq |φ(q)|2 exp [2πiq · (u − ri j )] 2 u i, j  sin (2πqri j ) 1  qi q j ∞ dq (1.2.84) = |φ(q)|2 π ri j 0 q i, j

since φ(q) is also spherically symmetric. Similarly, the self-energy is given by letting ri j → 0:   ∞ Ws = 2 qi2 |φ(q)|2 dq. (1.2.85) i

0

Hence the interaction energy is Wi =

  ∞ 1  |F(h)|2 2 |φ(h)| − 2 qi2 |φ(q)|2 dq, 2π V h2 0

(1.2.86)

i

h

from (1.2.83), and Wi =

   ∞ sin (2πqri j ) 1  qi q j ∞ dq − 2 |φ(q)|2 qi2 |φ(q)|2 dq π ri j 0 q 0 i, j

i

(1.2.87) from (1.2.84). Next suppose that σ (x) does not vanish outside the sphere whose radius is the minimum intercharge spacing. Then a term A, the correction due to overlap, must be added to the right-hand side of (1.2.86). However, from (1.2.84),  sin (2πqri j ) 1  qi q j ∞ dq + A, (1.2.88) |φ(q)|2 W = π ri j 0 q i, j

where

   1  qi q j 2 ∞ 2 sin (2πqri j ) A= 1− dq . |φ(q)| 2 ri j π 0 q i, j

Therefore we have   ∞ 1  qi q j 1  |F(h)|2 2 = |φ(h)| − 2 qi2 |φ(q)|2 dq 2 ri j 2π V h2 0 i, j i h    ∞  qi q j sin (2πqri j ) 2 1 1− dq . (1.2.89) + |φ(q)|2 2 ri j π 0 q i, j

1.2 Historical survey

25

The only conditions on φ are (1) φ(0) = 1; (2) |φ(q)|2 is integrable on [0, ∞]. The potential at x = xk is V (xk ) =

 ∞ ∂ Wi 1  Sk 2 = |φ(h)| − 4q |φ(q)|2 dq k ∂qk πV h2 0 h     qj 2 ∞ 2 sin 2πq|R j − xk | 1− dq , +2 |φ(q)| |R j − xk | π 0 q j

(1.2.90) where Sk =



qk  exp [2πih · (xk  − xk )].

(1.2.91)

k  ∈ cell

For example, if we take φ(h) = exp (−π h 2 /2K 2 ), (1.2.90) is precisely Ewald’s formula. Bertaut’s procedure was extended to multipole sums by Kanamori et al. [93]. An equivalent procedure was developed and extensively applied by Nijboer and de Wette [110], although actually it had been used several years earlier by  Placzek et al. [114]. The basic idea is that if S = n F(Rn ) converges slowly, then one should select a rapidly converging function φ(R) which is well behaved near R = 0 and write   F(Rn ) φ(Rn ) + F(Rn )[1 − φ(Rn )]. (1.2.92) S= n

n

The first series now converges well and the second can be transformed into a rapidly convergent form by Parseval’s theorem,    n ),  n ) G(K F(Rn ) G(Rn ) = (1.2.93) F(K n

n

 denotes the Fourier transform of F. As an example this was applied to where F  Slm (0|k, n) =

 ylm (θ, φ) exp (2πik · Rλ ) λ

Rλ2n+1

(1.2.94)

with φ(R) =

(n + l, π R 2 ) (n + l)

(1.2.95)

26

Lattice sums

to obtain  Slm (0|k, n)

 (n − l, π R 2 ) ylm (θ, φ) exp (2πik · Rλ ) 1 λ = (n + l) Rλ2n+1 λ

+

i l π 2n+l−3/2  |hλ − k|2n+l−3  λ

(−n +

3 2 , π |hλ





− k| ) ylm (θ , φ ) . 2

(1.2.96)

Here (θ  , φ  ) are the polar angles of h − k and  is the volume of the unit cell. In this way Kanamori et al. [93] evaluated several multipole terms of the electrostatic potential in a number of ionic lattices. In an interesting note Nijboer and de Wette [111] discussed the conditionally convergent series  P2 (cos θ ) (1.2.97) Rλ3 λ for a cubic lattice. This represents the field at the origin due to identical dipoles at all other lattice points and, by the Lorenz argument, should have the value − 13 π . However, if summed over successive shells the sum must vanish by symmetry. In keeping with the Lorenz ‘parallel capacitor’ argument, Nijboer and de Wette proposed that the series should be summed first over planes of dipoles and then over the various lattice planes. By separating out the summation over planes, which leads to a geometric sum, they applied their method to the two-dimensional planar sums and showed that, to computational accuracy, the value − 13 π is obtained. This planewise summation method was applied to calculating electrostatic potentials in metals by Sholl [126] and by McNeil [105]. A general discussion of the validity of resumming such conditionally convergent series was given by Campbell [31]. This planewise procedure was carried out on similar sums for a variety of lattices by de Wette [37] and de Wette and Schacher [38]. Nijboer’s and de Wette’s scheme √ was in turn generalized to the case of potentials f (R) for which f ( R)R −1/2 can be represented as a Laplace transform by Adler [1], who made use of the earlier work of Molière [107]. Adler gave explicit consideration to the sums   Ck f (|rkj |) Am ylm (θ kj , φ kj ) exp (iK · rkj ) (1.2.98) = j,k

m

which were required for studies of various optical properties of solids (see also Rudge [119]). Beginning about 1950, Emersleben published an extensive series of papers dealing with the electrostatic potentials of finite arrays of electric charges, [42, 43], but we shall not describe this work here since it bears only tangentially on our subject. In addition, Emersleben devoted a number of papers to various properties of Epstein zeta functions, [44, 45]. Since most of them deal

1.2 Historical survey

27

with functions of order 2, this work is subsumed and greatly generalized by more recent work to be described below. Of particular interest, however, is the pair of identities [45] (A)

(B)

  0 ··· 0  ··· Z  k1 =1 k P =1  h 1 + k1 /n . . . h P + k P /n    0 ··· 0    (s) = n P−s Z    n h , ..., n h  1 P n 

n 

  0  Z   x

    0 0  0    (s) + Z   (s)    1 1    y 2 −x 2 −y    0 0   1−1/2s  (s).  =2 Z   x+y x−y 

    (s)  

(1.2.99)

These are simply obtained as consequences of the cyclotomic equation (relating to sums over nth roots of unity) and represent analytic counterparts to the Hund–Naor relations, discussed below, which were obtained by simple physical arguments. Another significant development during the 1950s was the series of papers by Mackenzie [102], Benson [7], Benson and Schreiber [8], and Benson et al. [9] dealing with the practical evaluation of electrostatic sums, which were reduced to rapidly convergent series of modified Bessel or exponential functions. We shall illustrate their method by the sum S=

∞ 

[(k + a)2 + (l + b)2 ]−s .

(1.2.100)

k,l=−∞

The procedure is first to use (1.2.9) to exponentiate the summand, to apply Jacobi’s transformation in the form ∞  p=−∞

e

−( p+α)2 t

=

∞  π 1/2 

t

e−q

2 π 2 /t

cos (2qαπ ),

(1.2.101)

q=−∞

and then to perform the integral by means of Hobson’s representation for modified Bessel functions of the second kind,  ∞ 2 2 2 t β−1 e−k t−q π /t dt = 2(qπ/k)β K β (2π kq). (1.2.102) 0

28

Lattice sums

In this way we obtain the rapidly convergent series   ∞   q s−1/2   K s−1/2 [2π |q(l + b)|] cos (2qπa) l + b  q,l=−∞   ∞ 2π s   q s−1/2 K s−1/2 [2π |q(l + a)|] cos (2qπ b). = l + a  (s)

2π s S= (s)

(1.2.103)

q,l=−∞

Van der Hoff and Benson [134] provided a number of such representations for a variety of important electrostatic sums. It is interesting to note that, by applying this procedure to multiple sums whose values are known, the corresponding Bessel function series are obtained in closed form, as has been formalized by Hautot [73]. For example, applying this procedure to the Lorenz–Hardy series (Hardy [71]) discussed below, we obtain ∞ 

K 0 (mnπ ) = 14 [(4 −

√ 2)ζ ( 12 )β( 12 ) + ln (8π ) − γ ].

(1.2.104)

m,n=1

The following application is instructive and furnishes a very concise representation for the Madelung energy of the rock salt structure. We have μ=−

   l1

l2

(−1)l1 +l2 +l3 (l12 + l22 + l32 )−1/2 ,

(1.2.105)

l3

where the summation is over −∞ < li < ∞ and the prime denotes that l1 = l2 = l3 = 0 is omitted. Next we write, by symmetry, μ=−

   l1

= −3

l2

(−1)l1 +l2 +l3 (l12 + l22 + l32 )(l12 + l22 + l32 )−3/2

l3

 l1

l2

(−1)l1 l12 (l12 + l22 + l32 )−3/2 (−1)l2 +l3 ,

(1.2.106)

l3

and the prime can be omitted. By using (1.2.9) and (1.2.74) in the form ∞ 

(−1) exp (−l t) = l

l=−∞

2

 π 1/2  t

m odd

 −m 2 π 2 , exp 4t 

we find that  ∞ ∞ 12  l1 2 μ = −√ (−1) l1 exp (−l12 t) t 1/2 φ(t) dt, π 0 l1 =1

(1.2.107)

1.2 Historical survey where φ(t) =

29

 (−1)l2 +l3 exp [−(l22 + l32 )t] l2

l3



−(l22 + l32 )π 2 π  exp = . t l l 4t 2

(1.2.108)

3

odd

Next we note that

 ∞ (l22 + l32 )π 2 1/2 2 exp −l1 t − t dt 4t 0

1/2 π(l22 + l32 )1/2 =2 K 1/2 [πl1 (l12 + l22 )1/2 ] 2l1 = π 1/2l1−1 exp [−πl1 (l22 + l32 )1/2 ]

(1.2.109)

so that μ = −12π

∞   l1 =1 l2 ,l3 odd

l1 (−1)l1 exp [−πl1 (l22 + l32 )1/2 ].

(1.2.110)

Finally, summing the geometric l1 -series we obtain μ = 12π

 l2 ,l3 odd

= 12π

exp [−π(l22 + l32 )1/2 ] {1 + exp [−π(l22 + l32 )1/2 ]}2

∞ 

sech2

π 2

 (m 2 + n 2 )1/2 .

(1.2.111)

m,n=1 odd

This simple and rapidly convergent expression was discovered by Benson et al. [9] using a physical argument and was proved analytically by Mackenzie [101]. A related representation was worked out for the CsCl structure by Benson and van Zeggeren [10] and the original physical argument for (1.2.111) was extended to all rhombohedral lattices by Mackenzie [102]. A number of these results were obtained and generalized by Fumi and Tosi [53], who used Madelung’s method. In 1959 Maradudin and Weiss [104] devised an efficient numerical scheme for the sum  mn , (1.2.112) Ts = 2 (m + n 2 )s/2 m,n considered previously by Topping [132], which used the one-dimensional Poisson summation formula and asymptotic approximations for the incomplete gamma

30

Lattice sums

functions. The procedure appears to be useful for similar higher-dimensional sums but is too specialized for general application. The two-dimensional Poisson summation formula was the implicit basis for Madelung’s work in 1918 but was apparently not exploited more fully until the mid 1950s, when it was used explicitly by Nicholson [109] and Hove and Krumhansl [79] to rederive several older summations. We shall describe this procedure in more detail when we come to recent work by Pathria and Chaba [35], who have employed it extensively. Harris and Monkhorst [72] provided an interesting development of Nijboer’s and de Wette’s approach. They treated a given lattice as if it were composed of neutral cells made up of positive point charges embedded in a compensating uniform background of negative charge. By the use of Fourier integral representations they were able to represent the electrostatic potential at a point as a sum of two parts. One part depended on the lattice structure only and was a measure of the potential at a lattice point when we have unit positive charges at all lattice points balanced by negative charge distributed uniformly throughout the crystal. This term gave the electrostatic energy of the low-density plasma model for that given lattice. The second term depended both on the lattice structure and on the shape of the compensating charge distribution. If this balancing charge was replaced by localized point charges at appropriate points, this second term together with the first gave the Madelung energy of the lattice. Redlack and Grindlay [115, 116] in a series of papers considered the evaluation of Coulomb energies in finite size crystals of different shapes. They too were able to divide the potential into an intrinsic part, which was shape independent, and an extrinsic part, which was a function of shape and size. Their numerical evidence suggests that the intrinsic potential energy equals the Madelung energy of an infinite crystal. We believe that this summarizes the state of affairs up to approximately 1973. It is interesting to note that there was little attention given to the question of whether these important lattice sums might be obtained exactly in closed form. Starting in 1973 this question began to be considered in earnest and we deal with it in the next section.

1.3 The theta-function method in the analysis of lattice sums Theta functions have, of course, been used in the numerical evaluation of lattice sums since the work of Appell [2] and Ewald [49], and the Poisson summation formula so often invoked is just a particular case of a θ -function transformation. However, the analytic power of representing lattice sums as Mellin transforms of θ -functions (MTθ Fs) does not appear to have been utilized until 1973 when Glasser [55], [56] rediscovered and extended this approach. In order to understand the method, properties of θ -functions relevant to lattice sums will now be described.

1.3 The theta-function method in the analysis of lattice sums

31

Theta functions were first introduced by Jacobi [82] as a means of calculating elliptic functions. They are functions of a complex variable z and a parameter q. Our interest is in the case z = 0, so that for simplicity we consider θ -functions to depend on the single variable q. These functions have remarkable properties in that they may have both infinite series and infinite product representations; this leads to some astonishing identities among sums and products of θ -functions. Following Whittaker and Watson [139] we define the following quantities: θ2 = θ3 = θ4 =

∞ 

q (n−1/2) = 2q 1/4 (1 + q 2 + q 6 + · · · + q n(n+1) + · · · ), (1.3.1) 2

−∞ ∞  −∞ ∞ 

2

q n = 1 + 2q + 2q 2 + 2q 9 + · · · ,

(1.3.2)

2

(−1)n q n = 1 − 2q + 2q 2 − 2q 9 + · · · ,

−∞ ∞ 

θ1 = 2

(−1)n (2n + 1)q (n+1/2)

(1.3.3)

2

0

= 2q 1/4 (1 − 3q 2 + 5q 6 − 7q 12 + · · · ).

(1.3.4)

To these we add two series not given by Jacobi, θ5 = 2

∞ 

(−1)n q (2n−1/2) = 2q 1/4 (1 − q 2 − q 6 + q 12 + q 20 + · · · ), (1.3.5) 2

−∞

θ6 = 2q 1/4 (1 + 3q 2 − 5q 6 − 7q 12 + · · · ).

(1.3.6)

All these series can be expressed in terms of the infinite products   Q0 = ∞ (1 − q 2n ), Q1 = ∞ (1 + q 2n ), 1∞ 1∞ 2n−1 ), Q 3 = 1 (1 − q 2n−1 ), Q 2 = 1 (1 + q

(1.3.7)

viz., θ2 = 2q 1/4 Q 0 Q 21 ,

θ3 = Q 0 Q 22 ,

θ5 = 2q 1/4 Q 0 Q 1 /Q 2 (q 2 ),

θ4 = Q 0 Q 23 ,

θ1 = 2q 1/4 Q 30 ,

θ6 = 2q 1/4 Q 30 (q 2 )Q 32 (q 2 ).

(1.3.8) It is thus a matter of elementary algebra to obtain a large number of relations among the θ ’s; we list just a few to show the variety available. We have the additive relations θ3 + θ4 = 2θ3 (q 4 ),

θ3 − θ4 = 2θ2 (q 4 )

θ32 + θ42 = 2θ32 (q 2 ),

θ32 − θ42 = 2θ22 (q 2 )

θ34 = θ44 + θ24 .

(1.3.9)

32

Lattice sums

This last equation written in infinite product form, ∞

8 ∞

8 ∞

8    2n−1 2n−1 2n (1 + q ) = (1 − q ) + 16q (1 + q ) , (1.3.10) 1

1

1

led Jacobi [82] to describe it as ‘aequatio identica satis abstrusa’. It is easy to show that Q 1 Q 2 Q 3 = 1, Q 0 Q 3 = Q 0 (q 1/2 ), Q 1 Q 2 = Q(q 1/2 ), Q 2 Q 3 = Q 3 (q 2 ), Q 0 Q 1 = Q 0 (q 2 ), and these lead to multiplicative relations among the θ ’s, some examples of which are θ3 θ4 = θ42 (q 2 ),

2θ2 θ3 = θ22 (q 1/2 ),

2θ2 θ4 = θ52 (q 1/2 ), θ1

θ2 θ4 (q 4 ) = θ5 θ3 (q 2 ), θ6

= θ2 θ3 θ4 ,

=

θ5 θ32 (q 2 ).

(1.3.11) (1.3.12)

It is in these last two results that the power of θ -functions is best shown – namely the ability to reduce a product of multiple sums (in this case three) to just one sum. Nor are these the only examples. Jacobi gives many others by which squares, fourth powers, sixth powers, and even eighth powers of θ -functions may be contracted into single sums. Indeed, in certain combinations, any even power of θ -functions may be so contracted. Two examples are θ32

=1+4

∞  1

qn , 1 + q 2n

θ48

∞  n3 q n = 1 + 16 (−1)n . 1 − qn

(1.3.13)

1

A fairly large list of such relations was given by Zucker [142]. We now illustrate how to obtain lattice sums analytically using such relations. First define the Mellin transform Ms of f (t) by  ∞ (s)Ms [ f (t)] = t s−1 f (t) dt. 0

Let S(a, b, c) =



(al12 + bl1 l2 + cl22 )−s

(l1 ,l2 =0,0)

be a general two-dimensional sum. Then, e.g.,    S(1, 0, 1) = (l12 + l22 )−s = Ms exp [−(l12 + l22 )t] = Ms [θ32 − 1] (q = e−t ). Now, by using θ32 − 1 = 4

∞  1

∞

∞

1

0

 qn =4 q n (−1)m q 2nm , 2n 1+q

1.3 The theta-function method in the analysis of lattice sums

33

we have Ms [θ32 − 1] =

4 (s)

=4

∞ 



∞

t s−1

0

n −s

n=1

∞  ∞ 

(−1)m exp [−n(2m + 1)t] dt

n=1 m=0 ∞ 

(−1)m (2m + 1)−s = 4 ζ (s) β(s),

(1.3.14)

m=0

where β(s) =

∞ 

(−1)m (2m + 1)−s .

0

Thus, a two-dimensional sum has been evaluated exactly, i.e., we have expressed it as the product of simple sums. Such a result was first obtained by Lorenz [100]. Four steps are taken in this approach: (1) (2) (3) (4)

the sum is written as a Mellin transform; the transformed sum is written in terms of θ -functions of argument q = e−t ; the θ -functions series are transformed to other q-series where possible; the Mellin transform of the new series is re-evaluated.

We give another illustration with the eight-dimensional sum  (−1)l1 +l2 +···+l8 (l12 + l22 + · · · + l82 )−s =

Ms (θ48

− 1) = Ms 16

∞  (−1)n n 3 q n 1

= Ms 16

∞  ∞  1

η(s) =

1 − qn

(−1)n n 3 q n q nm = −16 ζ (s) η(s − 3),

(1.3.15)

1

∞ 

(−1)n+1 n −s .

1

When s = 4 this yields the attractive result that 

8π 4 ln 2 . (1.3.16) 45 Many similar examples are given by Glasser [56, 57] and Zucker [142]. The essential point of the four steps is whether we are able to find suitable expressions for the θ -function representation of a lattice sum. We can write quite generally  (m 2 + λn 2 )−s = Ms [θ3 θ3 (q λ ) − 1], (1.3.17) S(1, 0, λ) = (−1)l1 +l2 +···+l8 (l12 + l22 + · · · + l82 )−4 = −

but can proceed no further unless we have a q-series for θ3 θ3 (q λ ). Rather remarkably, many such series are available or may be constructed. If not, it

34

Lattice sums

appears that in other cases S(1, 0, λ) may be summed by number-theoretic techniques – see Section 1.4. This immediately poses the questions when such series may be constructed and whether it is possible to say when some general  S(a, b, c) =  (am 2 + bmn + cn 2 )−s may be summed exactly. It seems that, for two-dimensional sums, a criterion for solvability is available in the form of a conjecture by Glasser [57] as to when a two-dimensional sum is expressible in products of simple sums. This has been verified in all the cases known (Zucker and Robertson [147, 148]) and is discussed in more detail in Section 1.4. Here we shall focus attention on the physically more interesting threedimensional sums. Now, whereas in two dimensions it appears that we know precisely when we can express a lattice sum exactly, and also in four dimensions where a vast number of exact results are obtainable, in contrast, in three dimensions there is a paucity of exact results. Indeed, only two are known through the use of (1.3.12):  (−1)m [m 2 + n 2 + ( p − 12 )2 ]−s = Ms [θ2 θ3 θ4 ] = Ms [θ1 ] = 22s+1 β(2s − 1), (1.3.18)  m 2 2 −s 2 2  1 2 (−1) [(2m − 2 ) + 2n + 2 p ] = Ms [θ5 θ3 (q )] = Ms [θ6 ] = 22s+1 L −8 (2s − 1),

(1.3.19)

where L −8 (s) = 1 + 3−s − 5−s − 7−s · · · is an L-function (see Section 1.4). Neither of these results is of immediate physical interest; however, though θ -functions have not yet provided any physically important exact results in three dimensions, they still yield concise representations of lattice sums and their manipulation does away with the necessity of resorting to geometrical methods of handling three-dimensional sums. This process is seen at its best when applied to Hund’s procedure described above for calculating lattice potentials. By taking any lattice site as the origin of a coordinate system, Hund [81] set out to calculate the interaction of the origin particle with all other particles in a crystal. If there are several species of particle, each species being defined by a Bravais lattice spanned by the basis vectors a1 (i), a2 (i), a3 (i), one can place a particle of each species in turn at the origin and calculate its interaction with all other particles in terms of some characteristic length a, where, e.g., a = |a1 (i)|. This pure number is called the Hund potential of the ith species. By adding all the Hund potentials together and dividing by 2 (to allow for the fact that every interaction is counted twice) the total interaction energy of the crystal is expressed as a pure number in terms of the characteristic length. Hund was mainly concerned with ionic crystals for which the interaction is Coulombic, but his method is easily generalized. Let the coordinates of the nearest particle of species i to the origin be xa1 , ya2 , za3 , where x, y, z < 1. Let the origin particle interact with this species via a potential function F(r ), where r is given by r 2 = (m − x)2 a21 + (n − y)2 a22 + ( p −z)2 a23 +2(m − x)(n − y)a1 ·a2 +2(m − x)( p −z)a1 ·a3 +2(n − y)( p −z)a2 ·a3

1.3 The theta-function method in the analysis of lattice sums

35

and m, n, p are integers taking all values from −∞ to ∞ independently of one another. Then the Hund potential of the origin particle with this species is

 r  a12 a22 a32 2a1 · a2 2a2 · a3 2a1 · a3 . , , , , , , x, y, z : F = F H a a2 a2 a2 a2 a2 a2 m,n, p (1.3.20) The interaction of the origin particle with its own species corresponds to x = y = z = 0, and in the summation the point m = n = p = 0, which gives the interaction of the origin particle with itself, is excluded. It is quite straightforward to evaluate H for any structure and potential function. However, many simplifications may be made in cases of physical interest. Thus all species will generally be described by the same basic vectors a1 , a2 , a3 . Hund, in fact, concentrated on hexagonal and cubic structures. In the hexagonal case all lattice sites may be specified in terms of |a1 | = |a2 | = a, 2a1 · a2 = a 2 , a1 · a3 = a2 · a3 = 0, and |a3 | = ca, where c is called the axial ratio. For such crystals Hund evaluated some potentials for F(r ) = r −l , namely H (1, 1, 83 , 1, 0, 0, x, y, z : r −1 ) = φ(x, y, z)

(1.3.21)

H (1, 1, 4, 1, 0, 0, x, y, z : r −1 ) = X (x, y, z).

(1.3.22)

and

For cubic structures |a1 | = |a2 | = |a3 | = a; a1 · a2 = a2 · a3 = a3 · a1 = 0 and the Hund cubic potential is H (1, 1, 1, 0, 0, 0, x, y, z : r −1 ) = ψ(x, y, z : r −1 ).

(1.3.23)

Since most typical ionic crystals have cubic structures, we will concentrate on them. Thus ψ for particles interacting with an r −s potential may be written ψ(x, y, z : r −s ) =

∞   −∞ ∞ 

[(m − x)2 + (n − y)2 + ( p − z)2 ]−s

= Ms

−∞

q

(m−x)2

∞  −∞

q

(n−y)2

∞ 

q

( p−z)2

(1.3.24)

−∞

and, if x, y, z take on special values, we may express ψ in terms of θ -functions. Naor [108], in a little known but very important paper, fully developed Hund’s method to make a comprehensive study of cubic structures without free parameters. These cubic structures are those whose lattice sites lie at the intersection point of symmetry elements and are termed invariant cubic lattice complexes (ICLCs). For such structures (x, y, z) can only take on the values zero or multiples of 18 . Further, to preserve symmetry, a species may be required to occupy a particular set of sites simultaneously. Naor [108] then showed that only 17 potentials, which he denoted by capital roman letters, were required to calculate μ for all ICLCs. These

36

Lattice sums

Naor potentials were all various combinations of Hund potentials (see Table 1.1). By making use of identities determined by Hund from geometric considerations (see (1.2.67)–(1.2.69)), Naor found that only nine of these potentials were independent and conjectured that a linear combination of nine independent lattice sums was both sufficient and necessary to give the Madelung constant of any ionic crystal that was an ICLC. However, expressing ψ in terms of MTθ Fs not only makes the Hund identities obvious but furthermore shows that a hitherto unknown relation exists between two Naor potentials. ∞ (m−x)2 , which appears in the For example, consider the sum s(x) = −∞ q MTθ F representation of ψ. We have the following pairs of values: x = 0, s(0) = θ3 ;

x = 12 , s( 12 ) = θ2 ;

x = 14 , s( 14 ) = 2−1 θ2 (q 1/4 );

x = 18 , s( 18 ) = 2−2 [θ2 (q 1/16 ) + θ5 (q 1/16 )]; x = 38 , s( 38 ) = 2−2 [θ2 (q 1/16 ) − θ5 (q 1/16 )].

(1.3.25) = may be Furthermore s(x) = s(l − x). Hence expressed entirely in combinations of the θ -functions (1.3.1)–(1.3.6), and the full array of identities among them may be used to generate relations among the ψ. For example, ψ(0, 0, 0 : r −2s ) = Ms [θ33 − 1], where the 1 is subtracted in the transform to account for the omission of the term m = n = p = 0 in the original sum. Furthermore, ψ( 18 q , 18 q , 18 q )

ψ(0, 0, 12 : r −2s ) = Ms [θ32 θ2 ];

ψ( 18 q )

ψ(0, 12 , 12 : r −2s ) = Ms [θ3 θ22 ];

ψ( 12 , 12 , 12 : r −2s ) = Ms [θ23 ]. Now using the result θ3 = θ3 (q 4 ) + θ2 (q 4 ) plus the important property of Mellin transforms that Ms [ f (q k )] = k −s Ms [ f ],

(1.3.26)

we have ψ(0, 0, 0 : r −2s ) = Ms [θ33 − 1] = Ms {[θ3 (q 4 ) + θ2 (q 4 )]3 − 1} = 2−2s Ms [(θ3 + θ2 )3 − 1] = 2−2s Ms [θ33 − 1 + 3θ32 θ2 + 3θ3 θ22 + θ23 ] = 2−2s [ψ(0, 0, 0 : r −2s ) + 3ψ(0, 0, 12 : r −2s ) + 3ψ(0, 12 , 12 : r −2s ) + ψ( 12 , 12 , 12 : r −2s )],

(1.3.27)

and when s = 12 we have the Hund identity (1.2.69) without recourse to geometrical methods. Again, ψ( 14 , 14 , 14 : r −2s ) = 2−3 Ms [θ23 (q 1/4 )] = 22s−3 Ms [θ23 ] = 22s−3 ψ( 12 , 12 , 12 : r −2s )

Table 1.1 Naor and Hund potentials for the cubic lattices (k = 22s−3 )a Naor symbol

Hund potential

Theta function representation

Value for general s

Value for s = 12

Numerical value

a = k(a + 3b + 3c + d)

a(1) = b(1) + c(1) + d(1)/3

−2.837297479480619

A

ψ(0, 0, 0)

θ33 − 8 = k[(θ3 + θ4 )3 − 1]

B

ψ(0, 12 , 12 ) ψ(0, 0, 12 ) ψ( 12 , 12 , 12 ) ψ( 14 , 14 , 14 ) ψ( 14 , 14 , 12 ) 1 [ψ(0, 0, 1 ) 2 4 + ψ( 14 , 14 , 12 )] ψ(0, 14 , 14 ) ψ(0, 14 , 12 ) 1 [ψ( 1 , 1 , 1 ) 4 8 8 8 + 3ψ( 18 , 38 , 38 )] 1 [ψ( 3 , 3 , 3 ) 4 8 8 8 + 3ψ( 18 , 18 , 38 )] 1 [ψ(0, 1 , 1 ) 2 8 8 + ψ( 12 , 38 , 38 )]

θ3 θ22 = k[(θ3 + θ4 )(θ3 − θ4 )2 ] θ32 θ2 = k[(θ3 + θ4 )2 (θ3 − θ4 )] θ23 = k(θ3 − θ4 )3 2−3 θ23 (q 1/4 ) = k 2 (θ3 − θ4 )3 k[θ22 (θ3 − θ4 )]

G D E J K

Y Z L

M

N

k(a − b − c + d)

d(1)/3

−0.582521531544394

k(a + b − c − d)

[3b(1) − d(1)]/6

−0.095932304939804

k(a − 3b + 3c − d)

[3c(1) − a(1)]/2

−0.801935970028024

k 2 (a − 3b + 3c − d)

[3c(1) − a(1)]/8

−0.200483992507006

k 2 (a − b − c + d) − ke

[d(1) − 3e(1)]/12

+0.305934570506908

k[θ2 (θ32 + θ42 )]

k 2 (a + b − c − d) + k f

[3b(1) − d(1) + 6 f (1)]/24

−0.530673108834998

k[θ22 (θ3 + θ4 )]

k 2 (a − b − c + d) + ke

[d(1) + 3e(1)]/12

+0.239412342577787

k[θ2 (θ32 − θ42 )]

k 2 (a + b − c − d) − k f

[3b(1) − d(1) − 6 f (1)]/24

−0.353900722976810

k 2 [θ23 + θ53 ]

k 3 (a − 3b + 3c − d) + k 2 g

[3c(1) − a(1) + 2g(1)]/32

+0.108226339650319

k 2 [θ23 − θ53 ]

k 3 (a − 3b + 3c − d) − k 2 g

[3c(1) − a(1) − 2g(1)]/32

−0.208468335903822

k 2 [(θ3 + θ4 )(θ22 + θ52 )] + 22 k[θ2 θ3 θ4 ]

k 3 (a − b − c + d) + k 2 (e + h + j)

[d(1) + 3e(1) + 3h(1)

+ 25s−2 β(2s − 1)

+ 3 j (1)]/48 + 2−1/2

+1.074506464381240

Table 1.1 (cont.) Naor symbol P

R

S

X

T

Value for s = 12

Hund potential

Theta function representation

Value for general s

1 [ψ( 1 , 1 , 1 ) 2 2 2 8 + ψ(0, 38 , 38 )] 1 [ψ( 1 , 1 , 1 ) 2 4 8 8 + ψ( 14 , 38 , 38 )] 1 [ψ(0, 1 , 3 ) 2 8 8 + ψ( 12 , 18 , 38 )] ψ( 18 , 14 , 38 )

k 2 [(θ3 + θ4 )(θ22 + θ52 )]

k 3 (a − b − c + d) + k 2 (e + h + j)

[d(1) + 3e(1) + 3h(1)

− 22 k[θ2 θ3 θ4 ] k 2 [(θ3 − θ4 )(θ22 + θ52 )]

− 25s−2 β(2s − 1)

− 3 j (1)]/48 − 2−1/2

1 [ψ(0, 1 , 1 ) 4 8 4 + ψ(0, 14 , 38 ) + ψ( 12 , 18 , 14 ) + ψ( 12 , 14 , 38 )]

k 2 [(θ3 + θ4 )(θ22 − θ52 )]

k 3 (a − b − c + d)

[d(1) − 3e(1) + 3h(1)

+ k 2 (−e + h − j)

− 3 j (1)]/48

k 3 (a − b − c + d) + k 3 (e − h − j)

k 2 [θ2 (θ32 − θ42 )]

−0.339707097991855 +0.014147501743402

[d(1) + 3e(1) − 3h(1) −3 j (1)]/48

k 2 [(θ3 − θ4 )(θ22 − θ52 )]

Numerical value

−0.247693511905799

k 3 (a − b − c + d)

[d(1) − 3e(1) − 3h(1)

+ k 2 (−e − h + j)

+ 3 j (1)]/48

−0.279484055918394

k 3 (a + b − c − d) − k 3 f

[3b(1) − d(1) − 6 f (1)]/96

−0.088475180744203

a Hund’s identities: A = 3(B + G) + D, B = 2(Y + J ), D = 4E, E = 2(L + M), G = 2(Z + K ), J = 2(X + R), Y = N + P + 2S, Z = 4T, N − P = 22s−1 β(2s − 1).

1.3 The theta-function method in the analysis of lattice sums

39

and when s = 12 , 4ψ( 14 , 14 , 14 : r −1 ) = ψ( 12 , 12 , 12 : r −1 ), which is another of Hund’s identities. Using this approach Naor’s 17 potentials may be expressed in terms of MTθ Fs (Table 1.1). The Naor relations among the potentials are immediately apparent and the further relation N − P = 23s−2 Ms [θ1 ] = 25s−1 β(2s − 1)

(1.3.28) √ may be discovered. Thus for s = 12 and with β(0) = 12 we have N − P = 2, and only eight independent lattice sums at most are required to calculate μ for all ICLCs. Sakamoto [122] had noted this result from numerical work but did not give a proof. One cannot preclude the possibility that still further relations exist. The eight independent (as far as is known) sums chosen here to calculate μ are:  (−1)m (m 2 + n 2 + p 2 )−s = Ms [θ4 θ32 − 1], b = b(2s) =  (−1)m+n (m 2 + n 2 + p 2 )−s = Ms [θ42 θ3 − 1], c = c(2s) =  d = d(2s) = (−1)m+n+ p (m 2 + n 2 + p 2 )−s = Ms [θ43 − 1],  (−1)m [m 2 + (n − 12 )2 + ( p − 12 )2 ]−s = Ms [θ4 θ22 ], e = e(2s) =  (−1)m+n [m 2 + n 2 + ( p − 12 )2 ]−s = Ms [θ42 θ2 ], (1.3.29) f = f (2s) =  g = g(2s) = (−1)m+n+ p [(2m − 12 )2 + (2n − 12 )2 + (2 p − 12 )2 ]−s = Ms [θ53 ],  (−1)n+ p [m 2 + (2n − 12 )2 + (2 p − 12 )2 ]−s = Ms [θ3 θ52 ], h = h(2s) =  j = j (2s) = (−1)m+n+ p [m 2 + (2n − 12 )2 + (2 p − 12 )2 ]−s = Ms [θ4 θ52 ]. This choice is governed by the fact that each sum is convergent (albeit conditionally) for all s > 0, whereas some Hund potentials expressed as sums are divergent. For example, it is often convenient to use the self potential ψ(0, 0, 0 : r −2s ) =  2 (m + n 2 + p 2 )−s = Ms [θ33 − 1]. In its sum form a(2s) is a = a(2s) = divergent for s < 32 . But a(2s), which may be regarded as the three-dimensional analogue of ζ (2s), has only a simple pole at s = 32 and may be analytically continued for s < 32 by various functional relations. One such relation, which again uses θ -identities, is a(2s) = Ms [θ33 − 1] = Ms [ 18 (θ3 (q 1/4 ) + θ4 (q 1/4 ))3 − 1] = 22s−3 Ms [(θ3 + θ4 )3 − 8]. Hence a(2s) = 22s−3 [a(2s) + 3b(2s) + 3c(2s) + d(2s)]

(1.3.30)

and, in particular, a(1) = b(1) + c(1) + d(1)/3. Since b(1), c(l), d(l) are all convergent, we may give a value to a(1). Before considering these functional

40

Lattice sums

relations further it is instructive to show how the Hund–Naor method may be used for finding μ. As examples we will find μ for the NaCl, CsCl, and CuF2 structures. It is convenient to describe a species of particle in a cubic structure whose basic coordinates are (x, y, z) as a simple cubic lattice based on (x, y, z), SC(x, y, z) for short. Very often in cubic structures we find the same species occupying the four sites (x, y, z), (x, y + 12 , z + 12 ), (x + 12 , y, z + 12 ), (x + 12 , y + 12 , z); that is, the species forms a face-centred cubic lattice based on (x, y, z), FCC(x, y, z) for short. Many crystals are easily described in these terms; thus NaCl with Na+ as an origin may be described as Na+ FCC(0, 0, 0) + Cl− FCC( 12 , 12 , 12 ). Therefore, for Coulomb interactions, the potential for Na+ is ψ(Na+ ) = ψ(0, 0, 0) + 3ψ(0, 12 , 12 ) − ψ( 12 , 12 , 12 ) − 3ψ(0, 0, 12 ) and from Table 1.2 this is just 2d(1). If we now make Cl− the origin, the crystal is Cl− FCC(0, 0, 0) + Na+ FCC(0, 0, 0) and ψ(Cl− ) is also 2d(1). The total Coulomb energy is thus 12 [ψ(Na+ ) + ψ(Cl− )] = 2d(1), and this is μ(NaCl) by definition. Similarly, for CsCl, which is described by Cs+ SC(0, 0, 0) + Cl− SC( 12 , 12 , 12 ), we have μ(CsCl) = 32 b(1) + 12 d(1). The case of CaF2 is more complex. With Ca2+ as origin it is described by Ca2+ FC(0, 0, 0) + F− FCC( 14 , 14 , 14 ) + F− FCC( 34 , 34 , 34 ) and, therefore, ψ(Ca2+ ) = 4[ψ(0, 0, 0) + 3ψ(0, 12 , 12 )] − 16ψ( 14 , 14 , 14 ) = 6a(1) − 6c(1) + 4d(1). With F− as an origin, CaF2 is F− FCC(0, 0, 0) + F− FCC( 12 , 12 , 12 ) + Ca2+ FC( 14 , 14 , 14 ); thus ψ(F− ) = ψ(0, 0, 0) + 3ψ(0, 12 , 12 ) + ψ( 12 , 12 , 12 ) + 3ψ(0, 0, 12 ) −2 × 4ψ( 14 , 14 , 14 ) = 2a(1) − 3c(1). Hence μ(CaF2 ) = 12 [ψ(Ca2+ ) + 2ψ(F− )] = 6a(1) − 6c(1) + 2d(1) = 6b(1) + 4d(1). It is immediately evident that μ(CaF2 ) = 4μ(CsCl) + μ(NaCl).

(1.3.31)

This result or its equivalent was given first by P. Naor (unpublished) and by Bertaut [12], in 1954, and was derived in the above manner by Fumi and Tosi [52]. In a similar way it was shown that μ(CuO2 ) = 3b(1) + 3c(1) + 2d(1), and so, unless there is some unknown relation among b, c, and d, μ(CuO2 ) cannot be expressed in terms of μ(CsCl) and μ(NaCl). It is apparent, however, that because

1.3 The theta-function method in the analysis of lattice sums

41

Table 1.2 Madelung constants for a number of ionic crystals Crystal

b(1)

c(1)

NaCl

d(1)

e(1)

g(1)

μ

2

3.495129189

CsCl

3/2

1/2

2.035361510

ZnS

3/2

3/2

CaF2

6

4

11.63657523

Cu2 O

3

3

2

10.25945703

6

2

12.37746803

9/2

44.55497526

KZnF3 3/2

24

(A R = 1) √ (A R = 2)

1

1

(A R = 2)

1

1/2

6

24 24

8

49.50987213

3/2

24

25/2

58.53549202

LaAlO3 CaWO4

3.782926104

ReO3

2−1/2

BaTiO3 NaTaO3

2.254775948 2−1/2

2.282508448 −1/2

2.284666496

18

71.63183493

YOF

27/2

19/2

27.05607656

BiF3

6

10

22.12196280

1/2

10.91770035

BaLiF3

3/2

NbO

1/2

Li2 O

3/2

PdO

(AR =

√ 2)

10.77318449

1/2 3/2

6

6

[Pt(0, 0, 0)S( 41 , 14 , 14 )]

9/8

3/4

3/8

2.636813487

[Pt(0, 0, 0)S( 21 , 0, 0)]

1/2

1

1/2

2.741365176

[Pt(0, 0, 0)S( 21 , 0, 12 )]

1

3.649338711

1

1

−1/2

4.539442445

8

27.50917245

2.254775948

(CaFC(0, 0, 0), F I FC( 14 , 14 , 14 ), F I I FC( 21 , 0, 0))

3/2

11/2

10.77318449 63.49038889

K2 PtCl6

24

16

(NH4 )3 AlF6

24

10

6

33/2

75/2

CuVS4 Spinels

3.008539964

11/2

2

ZrI4

CaF2

3/2

3/2

(AR = 2) PtS

6

53.00500132 −6

66.7566350

Normal p = 2, q = 3 p = 3, q = 5/2 p = 4, q = 2

3

3

61

−6

128.5671125

27/4

27/4

221/4

−15/2

130.7743620

12

12

52

−8

138.1991295

Inverse p = 4, q = 2

4

4

62

−6

132.5694531

p = 2, q = 3

7

7

111/2

−15/2

131.7749472

42

Lattice sums

of the identities existing among Hund potentials, the number of independent sums required to obtain μ for different structures based on the same fundamental lattice will be small and we should not be surprised that relations such as (1.3.31) exist. What is surprising is that such relations were not discovered until 1954. Naor’s method was very complicated and used purely geometric techniques, by which a given structure was decomposed into a superposition of other structures, while Bertaut obtained his results by a procedure involving sums over the reciprocal lattice. As Tosi [133] points out, the Hund technique is straightforward and of general applicability whereas the other methods are liable to error. Indeed, Hund pointed out that Emersleben’s calculation of μ(Cu2 O) (Section 1.2) in Born potentials is incorrect, whereas it may be found in a few lines in Hund potentials. It is seen here that the sums b(1), c(1), and d(1) appear very often and, indeed, for the vast majority of the common ionic crystals only these three numbers are required to give μ. In Table 1.2 a list of such values is given. That just three numbers were sufficient was noted by Sakamoto [120], who gave tables of μ for many crystals in terms of three independent Born potentials. It is easy to transform from Born to Hund potentials and back again via the relation nπ

p n

=

n−1  n−1  n−1  q1 =q2 =q3 =0

ψ

q n

 exp

 −2πi(p · q) , n

(1.3.32)

but using Hund potentials to find μ is a simpler process. Sakamoto and his coworkers between 1953 and 1974 did a considerable amount of work on ionic crystals and it is worth summarizing the main points of their results here. Takahasi and Sakamoto [129] took up the problem of evaluating ψ to great accuracy, for many Naor potentials were unknown. Hund [81] had obtained a few to about three significant figures and Birman [15] added a few more. However, Takahasi and Sakamoto give ψ(q/4), 0 ≤ qi ≤ 2, to 15 places and ψ(q/6), 0 ≤ qi ≤ 3, to six places. Takahasi and Sakamoto also gave tables of ψ(q/24), 0 ≤ qi ≤ 12, to at least five places, and Sakamoto [121] in one of his papers on μ for crystals with quasi-cubic structures gave some values of ψ(q/n) with n = 960 and 1000 for several values of qi which he required. These values are, of course, unnecessary for ICLCs but are useful for crystals with cubic structures having free parameters. Sakamoto has also privately communicated to us ψ(q/8), 0 ≤ qi ≤ 4, to 15 places and thus has computed Naor’s 17 potentials for ICLC to this accuracy (see Table 1.1). All this numerical work was carried out by the Ewald method, which, though effective, is a comparatively slow method for computing lattice sums for Coulomb potentials. This method is still commonly used by workers evaluating lattice sums but more powerful methods are available, such as that of Van der Hoff and Benson [134] described above. (See also Graovac et al. [65] for an example of a different sort of sum.) The latter method was extended by Hautot [73], [74] to yield formulae for evaluating the better-known sums such as d(1) in terms of

1.3 The theta-function method in the analysis of lattice sums

43

elementary functions. So powerful are these techniques that these sums may be evaluated on a hand machine to many decimal places in a few minutes. The Van der Hoff–Benson–Hautot method makes essential use of the identities  ∞ ∞   25/2−s π s  2m + 1 s−1/2 (−1)n (m 2 + b2 )−s = (s) b −∞ m=0

× K s−1/2 [(2m + 1)bπ ], ∞ 

(m 2 + b2 )−s =

−∞

π 1/2 (s − b2s−1 (s)

1 2)

+ 25/2−s

s ≥ 0,  ∞  π s  2m s−1/2 (s)

m=1

× K s−1/2 [2mbπ ],

b s > 0, (1.3.33)

where K s is a modified Bessel function of the second kind (Watson [137]). Van der Hoff and Benson derived their results by putting the summand into integral form and, using the Poisson transformation formula ∞ ∞  π 1/2   2 2 2 e−n t = e−n π /t , (1.3.34) t −∞ −∞ transformed the sum into an integral representation for K s . Hautot used Schlomilch series with a Hankel transform to obtain equivalent results. The expansions are very rapidly convergent because the asymptotic behaviour of K s (z) is (π/2z)1/2 exp (−z). Since many two-dimensional sums are known exactly, (1.3.33) and (1.3.34) lead to interesting expressions for certain series of modified Bessel functions. For example, if in (1.3.33) we put b = n and sum over n, we have  ∞ ∞   25/2−s π s   2m + 1 s−1/2 n 2 2 −s (−1) (m + n ) = (s) n −∞ m=0

× K s−1/2 [(2m + 1)nπ ] = −4 × 2−s η(s) β(s),

η(s) = (1 − 21−s )ζ (s).

In particular, if s = 12 , it can be established that ∞  ∞  m=1 n=1

√ ln 2 2 − 2 1 + η( 2 ) β( 12 ). K 0 [(2m + 1)nπ ] = 4 4

(1.3.35)

Many similar results may be obtained (Chaba and Pathria [35]; Hautot [74]) and some are collected in Table 1.3. The application of (1.3.33) to evaluate d(1) where b2 = n 2 + p 2 led Hautot to the formula    d(1) = 2 ln 2 + 4 (−1)m+n K 0 [π(n 2 + p 2 )1/2 (2m + 1)], (1.3.36) which is essentially the result that Madelung [103] first obtained. However, no geometric considerations such as those described in Section 1.2 are required.

44

Lattice sums Table 1.3 Two-dimensional Bessel function sums ∞ 

K 0 (klπ ) =

k,l=1 odd ∞ 

∞ 

p=2 even

l=1 odd

∞  ∞ 

√ 4−3 2 1 ζ ( 2 ) β( 12 ) 8

K 0 ( plπ ) =

(−1)m−1 K 0 (mlπ ) =

m=1 l=1 odd ∞  ∞ 

K 0 (mlπ ) =

m=1 l=1 odd ∞ ∞  

K 0 (mpπ ) =

m=1 p=2 even ∞ 

K 0 (qpπ ), =

q, p=2 even ∞ 

K 0 (mnπ ) =

m,n=1 ∞  m,n=1 ∞ 

√ 2 1 1 ln 2 + ζ ( 2 ) β( 12 ) 4 8

K 0 (2π mn) =

√ 1− 2 1 1 ζ ( 2 ) β( 12 ) − ln 2 2 4

√ 2− 2 1 1 ln 2 + ζ ( 2 ) β( 12 ) 4 4 γ 1 1 ln (4π ) − + ζ ( 12 ) β( 12 ) 4 4 2

√ γ 4− 2 1 1 ln (2π ) − + ζ ( 2 ) β( 12 ) 4 4 8

√ γ 4− 2 1 1 ζ ( 2 ) β( 12 ) + ln (8π ) − 4 4 4 γ 1 1 1 ζ ( ) β( 12 ) + ln (4π ) − 2 2 4 4

√ γ 4− 2 1 1 ζ ( 2 ) β( 12 ) + ln (2π ) − 8 4 4 m,n=1

∞  ∞ 2  3 3 ) ζ ( 3 ) − 2π n 2 K 0 (2π mn) = β( 2 2 9 16π 2 K 0 (4π mn) =

m=1 n=1

Hautot [73] was also able to convert the Bessel function sum (1.3.36) into a sum of elementary functions by using the Poisson summation formula. This result, namely d(1) =

∞

∞

1

1

  (−1)m cosech(m 2 + p 2 )1/2 π 9 π − ln 2 + 12 , 2 2 (m 2 + p 2 )1/2

(1.3.37)

is by far the simplest formula for evaluating d(1). One requires only nine terms to give d(1) to 10 decimal places and its evaluation on an HP35 takes about a minute – an immense improvement on the traditional Ewald approach. However, the derivation of (1.3.37) by Hautot is not simple and furthermore (1.3.37) is just one of a large class of similar formulae, all of which may be derived in a fairly

1.3 The theta-function method in the analysis of lattice sums

45

elementary manner using MTθ Fs together with the Poisson summation formula. We will apply this method to the sums a(1), b(1), c(1), and d(1), which are of great importance in the evaluation of μ for ionic crystals (Zucker [143], [144]). First we point out that the Poisson summation formula is essentially a transformation formula for θ3 (Bellman [6]). In fact (1.3.34) may be written t 1/2 θ3 (e−π t ) = θ3 (e−π/t ).

(1.3.38)

In addition to this we have other Poisson-type formulae, such as t 1/2 θ2 (e−π/t ) = θ4 (e−π/t )

(2t)1/2 θ5 (e−π t ) = θ5 (e−π/4t ). (1.3.39) Now, by applying these results to MTθ Fs we are able to get functional relations between a(2s) and a(2s − 3) and similar ones for other lattice sums. Let us illustrate this for a one-dimensional sum.  Consider  (m 2 )−s = ζ (2s) = Ms [θ3 − 1]. Using (1.3.26) we transform this into Ms (θ3 − 1) = π s Ms [θ3 (q π ) − 1]. By applying the Poisson formula (1.3.38) we have and

Ms [θ3 − 1] = π s Ms [t −1/2 θ3 (e−π/t ) − 1].

(1.3.40)

Transforming from t to 1/t within the integral on the right-hand side of (1.3.40) we have Ms [θ3 − 1] =

π 2s−1 ( 12 − s) M1/2−s [θ3 − 1], (s)

(1.3.41)

π 2s−1 ( 12 − s) ζ (2s) = ζ (1 − 2s). (s) This is nothing other than the functional equation for the Riemann zeta function, and the above derivation is a standard method for obtaining this relation (Titchmarsh [131]). The structure of (1.3.41) is worth noting. The function ζ (2s) has a simple pole at s = 12 , so (1.3.41) is a reflection formula for ζ (2s) about the point s = 14 and we can thus calculate ζ (2s) for all s, both positive and negative, except at the pole. That a(2s) may be regarded as the three-dimensional analogue of ζ (2s) has already been pointed out. By applying a similar technique to a(2s), we obtain a(2s) =

π 2s−3/2 ( 32 − s) a(3 − 2s). (s)

(1.3.42)

The sum a(2s) has a simple pole at s = 32 and Equation (1.3.42) is a reflection formula for a(2s) about the point s = 34 ; we can calculate a(2s) for all s, both

46

Lattice sums

positive and negative, except at the pole. Now by using both (1.3.38) and (1.3.39) we can find similar functional relations for b(2s), c(2s), and d(2s): 22s b(2s) = K [a(3 − 2s) + b(3 − 2s) − c(3 − 2s) − d(3 − 2s)], 22s c(2s) = K [a(3 − 2s) − b(3 − 2s) − c(3 − 2s) + d(3 − 2s)], 22s d(2s) = K [a(3 − 2s) − 3b(3 − 2s) + 3c(3 − 2s) − d(3 − 2s)], (1.3.43) π 2s−3/2 ( 32 − s) K = . (s) If we put s =

1 2

in these results, we find the following interesting relations:

πa(1) = a(2),

π b(1) = 2b(2) + c(2), (1.3.44)

π c(1) = b(2) + c(2) + d(2),

π d(l) = 3c(2).

Furthermore, from (1.3.30) we have 3a(1) = 3b(1)+3c(1)+d(1)

a(2) = 3b(2)+3c(2)+d(2). (1.3.45)

and

Thus, of, the eight unknowns a(1)−d(1) and a(2)−d(2), only three are independent, say b(2), c(2), and d(2), and we should like to evaluate them. In their forms given by (1.3.29) they are hardly more rapidly convergent than b(1) − d(1), but by means of elementary formulas they may be transformed into very rapidly converging series. This will now be illustrated by evaluating d(1) from the relation π d(1) = 3c(2). The elementary summation formulas used are ∞  2π π 2 = M1 [θ3 e−x t ], (1.3.46) (m 2 + x 2 )−1 = + 2π x x x(e − 1) −∞ ∞  π cosech π x 2 = M1 [θ4 e−x t ], (−1)m (m 2 + x 2 )−1 = x −∞

(1.3.47)

∞  2π π 2 = M1 [θ2 e−x t ], (1.3.48) [(m − 12 )2 + x 2 ]−1 = − 2π x + 1) x x(e −∞

 Now, c(2) = (−1)m+n (m 2 + n 2 + p 2 )−1 = M1 [θ42 θ3 − 1] and we write  ∞ n 2 and T = n n2 θ3 = 1 + 2S and θ4 = 1 + 2T , where S = ∞ 1 q 1 (−1) q . Then, M1 [θ42 θ3 − 1] = M1 [θ4 (2T + 1)(2S + 1) − 1] = M1 [4θ4 ST + θ3 θ4 + θ42 − θ4 − 1] = M1 [θ3 θ4 − 1] + M1 [θ42 − 1] − M1 [θ4 − 1] ∞ ∞

 m −t (m 2 +n 2 ) + 4M1 θ4 (−1) e . (1.3.49) 1

1

1.3 The theta-function method in the analysis of lattice sums

47

The first three parts of (1.3.49) are easily evaluated by using the results of Glasser [56] and Zucker [58]; the last part is just (1.3.47) with x in the form of the sum of two squares. Therefore we have M1 [θ3 θ4 − 1] = − M1 [θ42 − 1] = −π ln 2,

π ln 2, 2

M1 [θ4 − 1] = −

π2 , 6

and c(2) = −

∞

∞

1

1

 3π ln 2 π 2 cosech (m 2 + n 2 )1/2 π + + 4π (−1)m . (1.3.50) 2 6 (m 2 + n 2 )1/2

Since d(1) = 3c(2)/π , we recover Hautot’s formula. We can rearrange the expression for c(2) as M1 [θ3 θ42 − 1], so M1 [θ3 θ42 − 1] = M1 [θ3 (2T + 1)2 − 1] = M1 [4θ3 T 2 + 2θ3 θ4 − θ3 − 1] = M1 [2θ3 θ4 − 2] − M1 [θ3 − 1] ∞ ∞

 2 2 + 4M1 θ3 (−1)m+n e−t (m +n ) . 1

1

Now, as before, M1 [2θ3 θ4 − 2] = −π ln 2, M1 [θ3 − 1] = π 2 /3 and using (1.3.46) we obtain 4M1 [θ3 T 2 ] = 4π

∞  ∞  1

+ 8π

1

(−1)m+n (m 2 + n 2 )1/2

∞  ∞  1

1

(−1)m+n . (m 2 + n 2 )1/2 [exp 2π(m 2 + n 2 )1/2 − 1]

The first double sum is easily evaluated (Glasser [56]), giving 4π [ln 2 − η( 12 ) β( 12 )], so finally we obtain √ d(1) = 9 ln 2 − π + 12( 2 − 1) ζ ( 12 ) β( 12 ) + 24

∞  ∞  1

1

(−1)m+n . (1.3.51) (m 2 + n 2 )1/2 [exp 2π(m 2 + n 2 )1/2 − 1]

This is another formula of Hautot [74] and Chaba and Pathria [34], and it requires only four terms in the double sum to give d(1) to 10 decimal places. However, the relations among θ -functions enable us to obtain many similar formulas without much effort. For example, θ3 θ4 = θ42 (q 2 ) so that c(2) = M1 [θ4 θ3 θ4 − 1] = M1 [θ4 θ42 (q 2 ) − 1] and, proceeding as before, we obtain

48

Lattice sums d(1) =

√ β π − √ ln [2( 2 − 1)] 2 2 ∞  ∞  (−1)m+n cosech [2m 2 + 2n 2 ]1/2 π + 12 . (2m 2 + 2n 2 )1/2 1

(1.3.52)

1

Further formulas are ∞  ∞  √ cosech (m 2 + 2n 2 )1/2 π d(1) = π − 3( 2 + 1) ln 2 + 24 (m 2 + 2n 2 )1/2 1

∞

∞

 cosech [2m(2n + 1)]1/2 π π = − − 12 (−1)m+n 2 [2m(2n + 1)]1/2 = −12π

1 1 ∞  ∞ 

sech2

1

(1.3.53)

1

1

π [(2m − 1)2 + (2n − 1)2 ]1/2 . 2

(1.3.54)

(1.3.55)

This last result is Mackenzie’s [102]. All these equations may be obtained by rearranging and transforming θ3 θ42 and all are formulas which lead to the evaluation of c(2) and d(1) in an extremely rapid way. This profusion of results is rather embarrassing as there appears to be no rhyme or reason in them. Without doubt there are many others of a similar nature and, in view of the later calculations in Section 1.5, this raises the question whether any of these formulae will lead to a closed form for d(1) in known transcendental constants. Certainly all the diagonal terms in the double sums in (1.3.50)–(1.3.55) may be evaluated. For example, if we put n = m in (1.3.52) we have 12π

∞  cosech 2mπ 1

2m

=π−

9 ln 2, 2

(1.3.56)

and this is the √ major contribution in the double sum. In fact we are able to sum ∞ 1 (cosech λnπ )/n in terms of π and logarithms of surds for any rational λ, so that a closed-form result for d(1) might yet be obtained. In a similar way expressions for b(2) and d(2) may be obtained and hence also for b(1) and c(1). Again, results for e(2) and h(2) may be found and these are related to e(1) and h(1) by the equations √ π e(1) = f (2), π f (1) = e(2), πg(1) = 2g(2), (1.3.57) π h(1) = h(2) + j (2), π j (1) = h(2) − j (2). In Table 1.4 we give some rapidly convergent double sums which can be used to evaluate b(2) and h(2). We emphasize again that these are not unique and many others may be found using θ -function relations. Nor are these methods restricted to s = 12 and 1. The intriguing formula 3b(3) + 3c(3) + d(3) = −4π ln 2

(1.3.58)

1.3 The theta-function method in the analysis of lattice sums

49

Table 1.4 Exponential and hyperbolic forms for the basic lattice sums with s = 1 b(2) =

√          √ √ 1 4π 2 √ 1 23 + 17 2 1/2 β − 2π − 2π 3 ln 2 − 4η − 2( 2 + 1) − 2π ln 2 2 3 2 ∞ ∞  √  + 8 2π 1

1

(−1)m+n √ (m 2 + n 2 )1/2 [exp 4 2π(m 2 + n 2 )1/2 − 1]

∞ ∞   √   π (−1)m+n cosech (2m 2 + 2n 2 )1/2 π π2 − √ ln 2 2 + 1 + 4π c(2) = 6 (2m 2 + 2n 2 )1/2 2 1 1

d(2) =

e(2) =

∞ ∞   √   cosech π(m 2 + 2n 2 )1/2 π2 π 2 + 1 + 8π (−1)n − π ln 2 − √ ln 2 3 (m 2 + 2n 2 )1/2 2 1 1

√ √ 2π ln (1 + 2) + 8π

∞ ∞   1

1

 1/2  1 2 cosech π 2m 2 + 2 n − 2 2 1/2  1 2m 2 + 2 n − 2

2 1/2  2+ n− 1 cosech π m ∞ ∞  2  √ √ f (2) = 2π ln (1 + 2) + 4π (−1)m 2 1/2  1 1 1 m2 + n − 2 

1/2  1 1 2 1 2 2n − ) + (m − ∞ ∞  2 2 2  g(2) = 2π (−1)n  2  2 1/2 −∞ −∞ 1 1 1 2n − + m− 2 2 2 ⎧ 2 1/2  ⎪ ⎪ ⎪ 2+ n− 1 ⎪ cosech π 2m ∞ ⎪ ∞  ⎨ 2  √ √ h(2) = π(4 − 2) ln (1 + 2) + 8π 2 1/2  ⎪ ⎪ 1 1 1 ⎪ ⎪ ⎪ 2m 2 + n − ⎩ 2 cosech π

 1/2 ⎫  ⎪ 1 2 ⎪ ⎪ ⎪ (−1)m cosech π 2m 2 + 2 n − ⎪ ⎬ 2 − 2 1/2  ⎪ ⎪ 1 ⎪ ⎪ ⎪ 2m 2 + 2 n − ⎭ 2 2 1/2  2 +2 n− 1 cosech π 2m (−1)m ∞ ∞   2 √ √ j (2) = 2π ln (1 + 2) + 8π  1/2  1 2 1 1 2m 2 + 2 n − 2

and results for larger-s sums may be obtained from higher-order analogues of (1.3.46)–(1.3.48). Thus, by differentiating (1.3.46) with respect to x we have ∞  −∞

(m 2 + x 2 )−2 =

π cosech2 π x π π + + , 2x 3 x 3 (e2π x − 1) 2x 2

(1.3.59)

50

Lattice sums

and so, for example, to evaluate a(4) = M2 [θ33 − 1] we find, following the procedure previously described, that  (m 2 + n 2 + p 2 )−2 a(4) = = 8ζ (2) β(2) − 2ζ (4) + 2π [ζ ( 32 ) β( 32 ) − ζ (3)] + 4π

∞  ∞ 

+ 4π 2

1 (m 2

1 1 ∞ ∞   1

1

+ n 2 )3/2 [exp 2π(m 2

+ n 2 )1/2 − 1]

cosech2 π(m 2 + n 2 )1/2 . 2(m 2 + n 2 )

(1.3.60)

The sum a(4) is one of the most slowly convergent sums, yet the above formula gives 10 significant figures if one retains only the first three terms from each double sum. A sequence of such formulas enables us to evaluate a(2s) and j (2s). These lattice sums are given in Table 1.5. We have dealt here only with cubic lattice sums, but the MTθ F method can be used for tetragonal and orthorhombic crystals (Zucker [143]), and hexagonal crystals can undoubtedly be handled this way though no analysis has yet been attempted. We conclude this section with a short account of some recent related work by Chaba and Pathria [32–35]. Though Chaba and Pathria do not explicitly make use of MTθ Fs, they use the Poisson summation formula to obtain many results similar to those described above. They consider n-dimensional sums of the form  exp (−a R 2 )/R 2s , (a)  cos (2π ε1l1 ) · · · cos (2π εn ln ) exp (−a R 2 )/R 2s , (1.3.61) (b)  exp [−a(l1 + ε1 )2 ] · · · exp [−a(ln + εn )2 ] (c) × [(l1 + ε1 )2 + · · · + (ln + εn )2 ]−s , where R 2 = l12 + l22 + · · · + ln2 . (A two-dimensional sum of this type was originally considered by Glasser [58].) Chaba and Pathria [32] then find a number of transformation formulas, e.g.,    exp (−a R 2 ) π 2 R2 π n/2 + (n−2)/2 E (4−n)/2 a R2 a 2 = π n/2 a (2−n)/2 + a + Cn (n = 1, 3, 4), n−2 = −π ln a + a + C2 (n = 2), (1.3.62) where



∞

Em = 1

t −m exp (−zt) dt,

m > 0.

(1.3.63)

1.3 The theta-function method in the analysis of lattice sums Table 1.5 Numerical values of the basic lattice sumsa s

a(2s)

b(2s)

c(2s)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

−2.8372974794(0) −8.913632917(0) ∞ 1.653231596(1) 1.037752483(1) 8.401920546(0) 7.467057780(0) 6.945807926(0) 6.628859200(0) 6.426119101(0) 6.292294498(0) 6.202149043(0) 6.140599581(0) 6.098184126(0) 6.068764295(0) 6.048263469(0)

−7.7438614142(−1) −3.013802247(−1) 2.2228710876(−1) 6.8922257409(−1) 1.0619823625(0) 1.3411086694(0) 1.5422045834(0) 1.6837514910(0) 1.7820445621(0) 1.8498105369(0) 1.8963845503(0) 1.9283773751(0) 1.9503780885(0) 1.9655380752(0) 1.9760103762(0) 1.9832637191(0)

−1.4803898065(0) −1.8300453641(0) −2.0461936439(0) −2.1568872986(0) −2.1968298536(0) −2.1955219241(0) −2.1738503493(0) −2.1448133764(0) −2.1155844408(0) −2.0895601364(0) −2.0679279901(0) −2.0507132697(0) −2.0374137084(0) −2.0273527819(0) −2.0198580752(0) −2.0143388512(0)

s

d(2s)

e(2s)

f (2s)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

−1.7475645946(0) −2.5193561521(0) −3.2386247661(0) −3.8631638072(0) −4.3786011499(0) −4.7884437139(0) −5.1054293720(0) −5.3455657733(0) −5.5246636380(0) −5.6566662484(0) −5.7530849044(0) −5.8230277889(0) −5.8734960409(0) −5.9097623766(0) −5.9357395661(0) −5.9542997599(0)

1.5401709012(0) 4.1458674219(0) 7.893704872(0) 1.3054752185(1) 2.0132691063(1) 2.9906494892(1) 4.3507844901(1) 6.2547665025(1) 8.9305752590(1) 1.2700438916(2) 1.8019768497(2) 2.5532313378(2) 3.6148194853(2) 5.1154270695(2) 7.2370214795(2) 1.0236924529(3)

1.3196705870(0) 4.8385895905(0) 1.2568251129(1) 2.8477747226(1) 6.0522064903(1) 1.2465784291(2) 2.5284972479(2) 5.0907147682(2) 1.0213048420(3) 2.0455377585(3) 4.0937626666(3) 8.1899751417(3) 1.6382172875(4) 3.2766354948(4) 6.5534521329(4) 1.3107067253(5)

s

g(2s)

h(2s)

j (2s)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

2.5335574044(0) 5.6281494825(0) 8.7850400797(0) 1.1857062518(1) 1.4885926393(1) 1.7985164284(1) 2.1283071340(1) 2.4902898412(1) 2.8960408839(1) 3.3567795759(1) 3.8839124460(1) 4.4895602845(1) 5.1870317651(1) 5.9912550183(1) 6.9191909161(1) 7.2368819704(2)

3.634989014(0) 7.7294890464(0) 1.2163847976(1) 1.7414440838(1) 2.4235087473(1) 3.3588428415(1) 4.6717592062(1) 6.5293708145(1) 9.1625607686(1) 1.2894706095(2) 1.8181430446(2) 2.5666225287(2) 3.6258741613(2) 5.1245294757(2) 7.2445017880(2) 1.0243062668(3)

1.2858465498(0) 3.6898829776(0) 7.3479876218(0) 1.2517226760(1) 1.9662159451(1) 2.9525897876(1) 4.3216738240(1) 6.2333952840(1) 8.9153618774(1) 1.2689861363(2) 1.8012547440(2) 2.5527454010(2) 3.6144961780(2) 5.1152139126(2) 7.2368819704(2) 1.0236833762(3)

a The values given must each be multiplied by 10n , where n is the number in

parentheses.

51

52

Lattice sums

By putting a = π , several interesting results may be obtained. It is shown that C1 = π 2 /3, that C2 = π(γ − η), where γ is Euler’s constant, and that η = ln [( 14 )4 ]/4π 3 . The constant C3 turns out to be equal to a(2), and C4 was shown by one of us to be −8 ln 2. This gives the attractive formula  exp [−π(l 2 + l 2 + l 2 + l 2 )] 1

2

3

4

l12 + l22 + l32 + l42

= π − 4 ln 2.

(1.3.64)

A sequence of such formulas was obtained, of which the most interesting are some exact results for double sums involving exponential functions similar to that which appears in (1.3.51). For example, ∞  ∞  1

1

+ n 2 )1/2 + (−1)m ] √ π 2− 2 1 9 η ln 2 − + ζ ( 2 ) β( 12 ) − , = 16 16 2 8 1 4 ( 4 ) η = ln 4π 3 1

(m 2

+ n 2 )1/2 [exp (m 2

(1.3.65)

(Chaba and Pathria [34]). Another exact result was derived by one of us from consideration of the dipole sum treated by Nijboer and de Wette [111]. This is ∞  ∞  1

1

1 (m 2 + n 2 )1/2 = 24 exp [2π(m 2 + n 2 )1/2 ] − 1



 ζ ( 3 ) β( 3 ) 1 −1 + 2 2 2 . π 8π

(1.3.66)

Formulae such as (1.3.65) and (1.3.66) give us hope that many three-dimensional sums may yet be summed exactly. Chaba and Pathria [35] also found several results similar to (1.3.35). In addition, they obtained some one-dimensional formulae similar to (1.3.65) and (1.3.66), such as ∞ 

m −1 (e2π m − 1)−1 =

1

π 1 1 ln 2 − − η, 2 12 4

(1.3.67)

but these results can, in fact, be generalized. In the notation of Whittaker and Watson [139], let K (k) be the complete elliptic integral of the first kind of modulus k:  K = 0

1

dx . [(1 − x 2 )(1 − k 2 x 2 )]1/2

With k 2 + k  2 = 1 and K (k  ) = K  , if (K  /K )2 = λ where λ, is rational, it is shown in Appendix 1.7 at the end of the chapter that our exact results for certain lattice sums permit us to express K in terms of gamma functions. By using these results the following sums can be evaluated for all rational λ (Zucker [146]):

1.3 The theta-function method in the analysis of lattice sums ∞ 

m

−1

1

53

√ √ 1 2K 3 kk  π λ −1 [exp (2π m λ) − 1] = − ln − 6 12 π3

√ ∞  √ k2 1 π λ m −1 −1 (−1) m [exp (2π m λ) − 1] = − ln − 12 16k  12 1

∞ 

m

1

−1

√ √ π λ 1 kK −1 + [exp (2π m λ) + 1] = ln 2 2π 4

√ (1.3.68) ∞  √ 2k 1 π λ m −1 −1 (−1) m [exp (2π m λ) + 1] = ln − 2 1 − k 4 1

∞  1

√ 1 2k  K (2m − 1)−1 {exp [(2 − 1)π λ] − 1}−1 = − ln 4 π

∞  √ 1 2K . (2m − 1)−1 {exp [(2 − 1)π λ] + 1}−1 = ln 4 π 1

Along with such sums, with each crystal lattice there is associated a class of multiple integrals related to the Green’s functions for a variety of physical processes. A subclass of these integrals, now called Watson integrals, described in Appendix 1.8, has been shown to be expressible in terms of K values for which λ is rational. Consequently there appears to be an intriguing connection between lattice sums and Watson integrals, whose elucidation could well have practical significance. In concluding this section, we should like to mention some recent work by Hall [68–70] and Barber [5] concerning the asymptotic behaviour of the lattice sums (1.3.61a). The asymptotic properties of these as a function of the parameter a have been studied. Hall found for two particular types of lattice sum that the asymptotic behaviour was analogous to that found in the theory of critical phenomena. In particular, the asymptotic behaviour was often independent of lattice structure, thus exhibiting a kind of universality. Barber [5] investigated sums of the form S(k, β) =



R−s f (k · R) e−a R

2

in the limit β → 0, |k| → 0 with β/|k| ∼ 1 and showed that these sums also exhibit crossover phenomena. The possible connection of lattice sums with critical phenomena should be worth exploring. The class of sums Sp =

 m,n

(−1)m+n+mn

exp [−(π/2)(m 2 + n 2 )] (m + in)2 p

54

Lattice sums

was studied by Boon and Zak [16]. These arise out of the completeness relation for coherent oscillator states on a von Neumann lattice and appear to be related to the invariants of Weierstrass elliptic functions. Some of their interesting properties are S1 = S3 = 0, S2 = − 14



S4 =

19 2 S , 14 2

(m + in)−4 = −

m,n

 8 ( 14 ) . 1024π 2

1.4 Number-theoretic approaches to lattice sums We pointed out earlier that Jones and Ingham [85] were the first to introduce number-theoretic considerations into the lattice sum problem. This idea was subsequently elaborated by Emersleben [46]. The principal aim, however, was to use number-theoretic tables, such as the representation functions for an integer as a sum of squares, as an aid in the numerical evaluation of sums. In this section we shall describe the application of number-theoretic ideas to the exact evaluation of a variety of multiple sums. In accordance with the exact results obtained already, it is reasonable to say that a lattice sum is solvable when it has been expressed in terms of a finite number of one-dimensional sums such as the Riemann zeta function. These are the Dirichlet L-series, and we shall begin with a brief survey of their more important properties. An elementary L-series (modulo k) is a function of the form ∞  χk (n) n −s , (1.4.1) L k (s, χ ) = n=1

where χk (a Dirichlet character) takes on the values ±1, 0 and has the properties χk (1) = 1,

χk (n + k) = χk (n),

χk (n) = 0

χk (nm) = χk (m) χk (n),

if k and n have a common factor d = ±1.

In particular, ζ (s) = L 1 (s, 1).

(1.4.2)

The most important properties of these functions were collected and proved by Zucker and Robertson [147]. For example, let P denote a square-free odd integer divisible by the prime p. Then (1) if k = 2P, P p , or 2α P (α > 3), there are no elementary L-series; (2) if k = P, 4P, there is just one such L-series; (3) if k = 8P, there are exactly two such series.

1.4 Number-theoretic approaches to lattice sums

55

Thus, L 3 = 1 − 2−s + 4−s − 5−s + · · · . One might suspect that the corresponding series with constant signs is an L-series for k = 3, but it is easily seen that this series is (1 − 3−s )L 1 . The two elementary series for k = 24 are L 24a = 1 + 5−s + 7−s + 11−s − 13−s − 17−s − 19−s − 23−s + · · · , L 24b = 1 + 5−s − 7−s − 11−s − 13−s − 17−s + 19−s + 23−s + · · · . (1.4.3) Since for the first series χk (k − 1) = −1, we denote it L −24 ; for a similar reason, the second series is denoted L 24 . Each of these L-series obeys a simple functional equation,   sin (sπ/2) L ±k (s) = C(s) L ±k (1 − s), C(s) = 2s π s−1 k −s+1/2 (1 − s), cos (sπ/2) (1.4.4) and can be summed at integer values:  (−1)m R(2m − 1)!/(2k)2m−1 , L ±k (1 − 2m) = 0,  0, (1.4.5) L ±k (−2m) = (−1)m R  (2m)!/(2k)2m , L k (2m) = Rk −1/2 π 2m ,

L −k (2m − 1) = R  k −1/2 π 2m−1 ,

where m is a positive integer and R, R  are rational numbers which depend on m, k. These Dirichlet series are intimately related to two-dimensional lattice sums of the form ∞   [Q(m, n)]−s = S(a, b, c; s), (1.4.6) m,n=−∞

where Q(m, n) = am 2 +bmn+cn 2 is an integer quadratic form. The discriminant of Q is the quantity d = b2 − 4ac, which we shall assume is negative so that Q never changes sign. A unimodular transformation is a change of variables m = αx + βy, n = γ x + δy, where α, β, γ , and δ are integers and αδ − βγ = 1. Under this substitution Q becomes a new form Q  (x, y), which is said to be equivalent to Q. It is easily verified that equivalent forms have the same discriminant, but it is also true that for each d < 0 there is a finite number h(d) of equivalence classes, called the class number of d. It is a remarkable result, proved by Dirichlet, that L −d (1) = 2π h(d)/w|d|1/2 , where ⎧ ⎪ d = −3a 2 , ⎪ 6, ⎨ (1.4.7) w= 4, d = −4a 2 , ⎪ ⎪ ⎩ 2 otherwise.

56

Lattice sums

(The number w is the number of substitutions which leave the quadratic form unchanged.) If we know the number of different solutions, R Q (N ), to the equation Q(m, n) = N then we can write (we assume that Q > 0) S(a, b, c; s) =

∞ 

R Q (N ) N −s .

(1.4.8)

N =1

It turns out in many cases, including a number not readily solvable through θ -function identities, that (1.4.8) can be decomposed into elementary Dirichlet series. We first require some additional number-theoretic tools. Two integers a, b leaving the same remainder when divided by n are said to be congruent modulo n: a ≡ b (mod n). A number congruent to a square is called a quadratic residue modulo n. Let p be a prime. Then the Legendre symbol is defined by ⎧ ⎪ 0, p divides m, ⎪ ⎨ (1.4.9) (m| p) = 1, m is a quadratic residue (mod p), ⎪ ⎪ ⎩ −1 otherwise. The generalization of this to composite moduli n = p1e1 · · · plel is called the Jacobi–Kronecker symbol and is defined by (m|n) = (m| p1 )e1 · · · (m| pl )el . In terms of this symbol we have L ±k (s) = Dirichlet’s formula Rd (k) = w(d)

∞ 1



(1.4.10)

(±k|n) n −s . Our principal tool is

(d|m)

(1.4.11)

m|k

for the number of ways in which an integer k is represented by a set of representations, one from each equivalence class, of the quadratic forms with discriminant d. Equation (1.4.11) is valid when k has no divisor in common with d (usually expressed as (k, d) = 1) and the sum is over all the divisors of k. The solvability of S(a, b, c; s) devolves upon the ability to extract from Rd (k) the representation function R Q (k) itself, and there is a classification of forms which suggests when this is possible. If there is only one class of forms with discriminant d, in which case it is convenient to consider the reduced form, for which b ≤ a ≤ c, (1.4.11) gives R Q (k) directly for those k which are relatively prime to d ((k, d) = 1). For other values of k, R Q (k) can be obtained ad hoc. We shall illustrate this by the lattice sums for square and triangular lattices:

1.4 Number-theoretic approaches to lattice sums S(1, 0, 1; s) = S(1, 1, 1; s) =

 

(m 2 + n 2 )−s ,

Q1 = m 2 + n2,

57

d = −4, (1.4.12)

2 −s

(m + mn + n ) , 2

Q 2 = m + mn + n , 2

2

d = −3.

When k is odd it can have no factor in common with d = −4 so, by Dirichlet’s formula,  (−4|m). R Q 1 (k) = 4 m|k

If k = 2q then if 2q = x 2 + y 2 we can write x + y = 2a, x − y = 2b, so q = a 2 + b2 is also of this form, that is,  (−1|m). R Q 1 (2l k) = R Q 1 (k) = 4 m|k odd

Hence ∞ ∞ ∞   R Q 1 (n) R Q 1 (k) 1  R Q 1 (k) S(1, 0, 1; s) = = + s + ··· ns ks 2 k=1 ks k=1 m=1

odd

= (1 − 2−s )−1

∞ 

odd

R Q 1 (k) k −s

(1.4.13)

k=1 odd

= 4(1 − 2−s )

∞   k=1 odd

(−4|m) k −s .

m|k

However, we can write k = mr and sum freely over the odd values of m and r . Therefore, S(1, 0, 1; s) = 4(1 − 2−s )−1

∞ 

(−4|m) m −s

m=1 odd

= 4L −4 (s) L 1 (s) = 4 β(s) ζ (s),

∞ 

r −s

r =1 odd

(1.4.14)

since 1 + 1/3s + 1/5s + · · · = ζ (s) − (1/2s + 1/4s + · · · ) = ζ (s)(1 − 2−s ) and L −4 (s) = 1 − 3−s + 5−s − 7−s + · · · ≡ β(s). Similarly, when (k, 3) = 1,  (−3|m). (1.4.15) R Q 2 (k) = 6 m|k

58

Lattice sums

When 3k = m 2 + mn + n 2 , then 3 divides 3k + 3(mn + n 2 ) = (m + 2n)2 . Thus m + 2n = 3a and m − n = 3a − 3n = 3b. Consequently, 3k = 3(a 2 + ab + b2 ) so k is also represented by Q 2 . This means that R Q 2 (3l k) = R Q 2 (k). Following the argument above, we have S(1, 1, 1; s) = 6 L 1 (s) L −3 (s) = 6 ζ (s) g(s),

(1.4.16)

where g(s) = 1 − 2−s + 4−s − 5−s + 7−s . . . More generally it can happen that the reduced forms in the various h(d) equivalence classes represent disjoint sets of integers, which in number-theoretic terms is expressed by saying that there is one class per genus. In this case there exist functions taking on different values on the sets of integers, with whose help it is possible to disentangle the R Q (n) from the Rd (n) as we shall illustrate with the sum S(1, 0, 16; s). The discriminant in this case is −64, for which the class number is 2, and so we must deal with the two reduced forms Q 1 = m 2 + 16n 2 ,

Q 2 = 4m 2 + 4mn + 5n 2 .

(A convenient tabulation of these facts is given in L. E. Dickson [39].) An exam√ ination of the character table for the Abelian group of order 8 = −d yields the function ⎧ ⎪ 0, n even, ⎪ ⎨ (1.4.17) C(n) = 1, n = ±1 (mod 8), ⎪ ⎪ ⎩ −1, n = ±3 (mod 8). It is easily verified that this has the property  C(k) =

1,

k = Q 1 (m, n) odd,

−1,

k = Q 2 (m, n) odd.

Hence when k is odd we find from Dirichlet’s formula that  R Q 1 (k) = [C(k) + 1] (−64|m).

(1.4.18)

(1.4.19)

m|k

When 2k = x 2 + 16y 2 , k is of the form 2(a 2 + 4y 2 ), which is impossible. Hence R Q 1 (2k) = 0. If 4k = x 2 + 16y 2 then k must have the form a 2 + 4y 2 . The representation function for this quadratic form can be worked out as in the preceding paragraph, and we obtain  (−16|m). R Q 1 (4k) = 2 m|k

1.4 Number-theoretic approaches to lattice sums

59

When 8k = x 2 + 16y 2 , k has the form 2(a 2 + y 2 ) so again R Q 1 (8k) = 0. Finally, when 2l k = x 2 + 16y 2 , l ≥ 4, factoring out factors of 2 leads again to the two-squares problem and  R Q 1 (2l k) = 4 (−4|m). m|k

This specifies R Q 1 (n) completely and the summation procedure used above yields S(1, 0, 16; s) = (1 − 2−s + 21−2s − 21−3s + 22−4s )L 1 (s) L −4 (s) +L 8 (s) L −8 (s).

(1.4.20)

This procedure was applied systematically by Zucker and Robertson [148] and leads to the results in Table 1.6. When there is more than one class per genus, that is, when the reduced forms corresponding to the discriminant d represent integers in common, the situation is exceedingly complex. Indeed, we conjecture (Glasser [57]) that only lattice sums which correspond to discriminants having one class per genus are solvable. This has been verified for discriminants |d|  40000 and so far no lattice sum outside this class has been solved. The proof of this conjecture remains an interesting open question. In addition to S(a, b, c; s) it is also desirable to consider the alternating sums  (−1)m Q −s , S1 (a, b, c; s) = S2 (a, b, c; s) = S1,2 (a, b, c; s) =

 

(−1)n Q −s ,

(1.4.21)

(−1)m+n Q −s .

There appears to be no systematic approach to these, but they are not independent of the basic sum S and the following relations have been derived (the details can be found in Zucker and Robertson [147, 148]): S(1, 0, λ; s) + S1 (1, 0, λ; s) + S2 (1, 0, λ; s) + S1,2 (1, 0, λ; s) = 22−2s S(1, 0, λ; s), S(1, 0, λ; s) + S2 (1, 0, λ; s) = 2 S(1, 0, 4λ; s), S(1, 1, λ ; s) + S2 (1, 1, λ ; s) = 2 S(1, 0, λ; s)

(λ = 14 (1 + λ)),

S1,2 (1, 1, λ ; s) = S1 (1, 1, λ ; s) = S1,2 (1, 0, λ; s), S(1, 1, λ ; s) + S1 (1, 1, λ ; s) + S2 (1, 1, λ ; s) + S1,2 (1, 1, λ ; s) = 22−2s S(1, 1, λ ; s). (1.4.22)

60

Lattice sums

Table 1.6 Principal solutions of all solvable S(a, b, c; s) for various c-values and (i) a = 1, b = 0, (ii) a = 1, b = 1 c

(i) a = 1, b = 0

1 2 3 4 5 6 7 8 9 10 12 13 15 16 18 21 22 24 25 28 30 33 37 40 42 45

4L +1 L −4 2L +1 L −8 2(1 + 21−2s )L +1 L −3 2(1 − 2−s + 21−2s )L +1 L −4 L +1 L −20 + L −4 L +5 L +1 L −24 + L −3 L +8 2(1 − 21−s + 21−2s )L +1 L −7 (1 − 2−s + 21−2s )L +1 L −8 + L −4 L +8 (1 + 31−2s )L +1 L −4 + L −3 L +12 L +1 L −40 + L +5 L −8 (1 − 2−2s + 22−4s )L +1 L −3 + L −4 L +12 L +1 L −52 + L −4 L +13 (1 − 21−s + 21−2s )L +1 L −15 + (1 + 21−s + 21−2s )L −3 L +5 (1 − 2−s + 21−2s − 21−3s + 22−4s )L +1 L −4 + L −8 L +8 (1 − 2 × 3−s + 31−2s )L +1 L −8 + L −3 L +24 2−1 (L +1 L −84 + L −3 L +28 + L −7 L +12 + L −4 L +21 ) L +1 L −88 + L −11 L +8 2−1 [(1 − 2−s + 21−2s )L +1 L −24 + (1 + 2−s + 21−2s )L −3 L +8 + L −4 L +24 + L −8 L +12 ] (1 − 2.5−s + 51−2s )L +1 L −4 + L +5 L −20 (1 − 21−s + 3 × 2−2s − 22−3s + 22−4s )L +1 L −7 + L −4 L +28 2−1 [L +1 L −120 + L −3 L +40 + L +5 L −24 + L +8 L −15 ] 2−1 [L +1 L −132 + L −3 L +44 + L −11 L +12 + L −4 L +33 ] L +1 L −148 + L −4 L +37 2−1 [(1 − 2−s + 21−2s )L +1 L −40 + (1 + 2−s + 21−2s )L +5 L −8 + L −4 L +40 + L −8 L +20 ] 2−1 [L +1 L −168 + L −3 L +56 + L −7 L +24 + L −8 L +21 ] 2−1 [(1 − 2 × 3−s + 31−2s )L +1 L −20 + (1 + 2 × 3−s + 31−2s )L −4 L +15 + L −3 L +60 + L +12 L −15 ] 2−1 [(1 + 2−2s + 21−4s + 23−6s )L +1 L −3 + (1 + 21−2s )L −4 L +12 + L −8 L +24 + L +8 L −24 ] 2−1 [L +1 L −228 + L −3 L +76 + L −19 L +12 + L −4 L +57 ] L +1 L −232 + L −8 L +29 2−1 [(1 − 21−s + 3 × 2−2s − 22−3s + 22−4s )L +1 L −15 + L −4 L +60 + (1 + 21−s + 3 × 2−2s + 22−3s + 22−4s )L −3 L +5 + L +12 L −20 ] 2−1 [L +1 L −180 + L +5 L −56 + L −7 L +40 + L +8 L −35 ] 2−1 [(1 − 2.3−s + 31−2s )(1 − 2−s + 21−2s )L +1 L −8 + L +12 L −24 + (1 + 2 × 3−s + 31−2s )L −4 L +8 + (1 + 2−s + 21−2s )L −3 L +24 ] 2−1 [L +1 L −312 + L −3 L +104 + L +13 L −24 + L +8 L −39 ] 2−1 [L +1 L −340 + L +5 L −68 + L −20 + L +17 + L −4 L +85 ] 2−1 [(1 − 2−s + 21−2s )L +1 L −88 + (1 + 2−s + 21−2s )L −11 L +8 + L −4 L +88 + L −8 L +44 ] 2−1 [L +1 L −372 + L −3 L +124 + L −31 L +12 + L −4 L +93 ] 2−1 [L +1 L −408 + L −3 L +136 + L +17 L −24 + L +8 L −51 ] 2−2 [L +1 L −420 + L −3 L +140 + L +5 L −84 + L −7 L +60 + L −15 L +28 + L +21 L −20 + L −35 L +12 + L −4 L +105 ] 2−1 [(1 − 21−s + 3 × 2−2s − 22−3s + 3 × 21−4s − 23−5s + 23−6s )L +1 L −7 + (1 + 21−2s )L −4 L +28 + L −8 L +56 + L +8 L −56 ] 2−2 [(1 − 2−s + 21−2s )L +1 L −120 + L −4 L +120 + (1 + 2−s + 21−2s )L −3 L +40 + L +12 L −40 + (1 + 2−s + 21−2s )L +5 L −24 + L −20 L +24 + (1 − 2−s + 21−2s )L −15 L +8 + L −8 L +60 2−1 [L +1 L −520 + L +5 L −104 + L +13 L −40 + L −8 L +65 ] 2−1 [L +1 L −532 + L −7 L +76 + L −19 L +28 + L −4 L +133 ] 2−2 [L +1 L −660 + L +5 L −132 + L −3 L +220 + L −11 L +60 + L −15 L +44 + L +33 L −20 + L −55 L +12 + L −4 L +165 ]

48 57 58 60 70 72 78 85 88 93 102 105 112 120

130 133 165

1.4 Number-theoretic approaches to lattice sums

61

Table 1.6 (cont.) c

(i) a = 1, b = 0

168

2−2 [(1 − 2−s + 21−2s )L +1 L −168 + L −4 L +168 + (1 + 2−s + 21−2s )L −3 L +56 + L +12 L −56 + (1 − 2−s + 21−2s )L −7 L +24 + L +28 L −24 + (1 + 2−s + 21−2s ) × L −8 L +21 + L +8 L −84 ] 2−1 [L +1 L −708 + L −3 L +236 + L −59 L +12 + L −4 L +177 ] 2−1 [L +1 L −760 + L +5 L −152 + L −19 L +40 + L +8 L −95 ] 2−2 [L +1 L −840 + L −3 L +280 + L +5 L −168 + L −7 L +120 + L −15 L +56 + L +21 L −40 + L −35 L +24 + L −8 L +105 ] 2−1 [(1 − 2−s + 21−2s )L +1 L −232 + L −4 L +232 + (1 + 2−s + 21−2s )L −8 L +29 + L +8 L −116 ] 2−2 [(1 − 21−s + 3 × 2−2s − 22−3s + 3 × 21−4s − 22−5s + 22−6s )L +1 L −15 + (1+21−s + 3 × 2−2s + 22−3s + 3 × 21−4s + 22−5s + 22−6s )L −3 L +5 + (1 + 21−2s ) × L −4 L +60 + (1 + 21−2s )L +12 L −20 + L −24 L +40 + L +24 L −40 + L −8 L +120 + L +8 L −120 ] 2−1 [L +1 L −1012 + L −11 L +92 + L −23 L +44 + L −4 L +253 ] 2−2 [L +1 L −1092 + L −3 L +364 + L −7 L +156 + L +13 L −84 + L +21 L −52 + L −39 L +28 + L −91 L +12 + L −4 L +273 ] 2−2 [(1 − 2−s + 21−2s )L +1 L −280 + L −4 L +280 + (1 + 2−s + 21−2s )L +5 L −56 + L −20 L +56 + (1 − 2−s + 21−2s )L −7 L +40 + L +28 L −40 + (1 + 2−s + 21−2s ) × L −35 L +8 + L −8 L +140 ] 2−2 [(1 − 2−s + 21−2s )L +1 L −312 + L −4 L +312 + (1 + 2−s + 21−2s )L −3 L +104 + L +12 L −104 + (1 + 2−s + 21−2s )L +13 L −24 + L −52 L +24 + (1 − 2−s + 21−2s ) × L −39 L +8 + L −8 L +156 ] 2−2 [L +1 L −1320 + L −3 L +440 + L +5 L −264 + L −11 L +120 + L −15 L +88 + L +33 L −40 + L −55 L +24 + L −8 L +165 ] 2−2 [L +1 L −1380 + L −3 L +460 + L +5 L −276 + L −23 L +60 + L −15 L +92 + L +69 L −20 + L −115 L +12 + L −4 L +345 ] 2−2 [L +1 L −1428 + L −3 L +476 + L −7 L +204 + L +17 L −84 + L +21 L −68 + L −51 L +28 + L −119 L +12 + L −4 L +357 ] 2−2 [L +1 L −1540 + L +5 L −308 + L −7 L +220 + L −11 L +140 + L −55 L +28 + L −35 L +44 + L +77 L −20 + L −4 L +385 ] 2−2 [(1 − 2−s + 21−2s )L +1 L −408 + L −4 L +408 + (1 + 2−s + 21−2s )L −3 L +136 + L +12 L −136 + (1 − 2−s + 21−2s )L +17 L −24 + L −68 L +24 + (1 + 2−s + 21−2s ) × L −51 L +8 + L −8 L +204 ] 2−2 [L +1 L −1848 + L −3 L +616 + L −7 L +264 + L −11 L +168 + L +21 L −88 + L +33 L −56 + L +77 L −24 + L +8 L −231 ] 2−2 [(1 − 2−s + 21−2s )L +1 L −520 + L −4 L +520 + (1 + 2−s + 21−2s )L +5 L −104 + L −20 L +104 + (1 + 2−s + 21−2s )L +13 L −40 + L −52 L +40 + (1 − 2−s + 21−2s ) × L −8 L +65 + L +8 L −260 ] 2−2 [(1 − 2−s + 21−2s )L +1 L −760 + L −4 L +760 + (1 + 2−s + 21−2s )L +5 L −162 + L −20 L +162 + (1 + 2−s + 21−2s )L −19 L +40 + L +76 L −40 + (1 − 2−s + 21−2s ) × L −95 L +8 + L −8 L +380 ] 2−3 [(1 − 2−s + 21−2s )L +1 L −840 + L −4 L +840 + (1 + 2−s + 21−2s )L −3 L +280 + L +12 L −280 + (1 + 2−s + 21−2s )L +5 L −168 + L −20 L +168 + (1 − 2−s + 21−2s ) × L −7 L +120 + L +28 L −120 + (1 − 2−s + 21−2s )L −15 L +56 + L +60 L −56 + (1 + 2−s + 21−2s )L +21 L −40 + L −84 L +40 + (1 + 2−s + 21−2s )L −35 L +24 + L +140 L −24 + (1 − 2−s + 21−2s )L +105 L +8 + L −420 L +8 ] −3 2 [(1 − 2−s + 21−2s )L +1 L −1320 + L −4 L +1320 + (1 + 2−s + 21−2s )L −3 L +440 + L +12 L −440 + (1 + 2−s + 21−2s )L +5 L −264 + L −20 L +264 + (1 + 2−s + 21−2s ) × L −11 L +120 + L +44 L −120 + (1 − 2−s + 21−2s )L −15 L +88 + L +60 L −88 + (1 − 2−s + 21−2s )L +33 L −40 + L −132 L +40 + (1 − 2−s + 21−2s )L −55 L +24 + L +220 L −24 + (1 + 2−s + 21−2s )L +165 L −8 + L +8 L −660 ]

177 190 210 232 240

253 273 280

312

330 345 357 385 408

462 520

760

840

1320

62

Lattice sums Table 1.6 (cont.)

c

(i) a = 1, b = 0

1365

2−3 [L +1 L −5460 + L −3 L +1820 + L +5 L −1092 + L −7 L +780 + L +13 L −420 + L −15 L +364 + L +21 L −260 + L −35 L +156 + L −39 L +140 + L +65 L −84 + L −91 L +60 + L +105 L −52 + L −195 L +28 + L +273 L −20 + L −455 L +12 + L −4 L +1365 ] 2−3 [(1 − 2−s + 21−2s )L +1 L −1848 + L −4 L +1848 + (1 + 2−s + 21−2s )L −3 L +616 + L +12 L −616 + (1 − 2−s + 21−2s )L −7 L +264 + L +28 L −264 + (1 + 2−s + 21−2s ) × L −11 L +168 + L +44 L −168 + (1 + 2−s + 21−2s )L +21 L −88 + L −84 L +88 + (1 − 2−s + 21−2s )L +33 L −56 + L −132 L +56 + (1 + 2−s + 21−2s )L +77 L −24 + L −308 L +24 + (1 − 2−s + 21−2s )L −231 L +8 + L −8 L +924 ]

1848

c

(ii) a = 1, b = 1

1 2 3 4 5 7 9 11 13 17 19 23 25 29 31 37 41 47 49 59 67 79

6L +1 L −3 2L +1 L −7 2L +1 L −11 L +1 L −15 + L −3 L +5 2L +1 L −19 2(1 − 2 × 3−s + 31−2s )L +1 L −3 L +1 L −35 + L +5 L −7 2L +1 L −43 L +1 L −51 + L −3 L +17 2L +1 L −67 (1 + 51−2s )L +1 L −3 + L +5 L −15 L +1 L −91 + L −7 L +13 (1 − 2 × 3−s + 31−2s )L +1 L −11 + L −3 L +33 L +1 L −115 + L +5 L −23 L +1 L −123 + L −3 L +41 (1 − 2 × 7−s + 71−2s )L +1 L −3 + L −7 L +21 2L +1 L −163 L +1 L −187 + L −11 L +17 2−1 [L +1 L −195 + L −3 L +65 + L +5 L −33 + L −15 L +13 ] L +1 L −235 + L +5 L −47 L +1 L −267 + L −3 L +89 2−1 [(1 − 2 × 3−s + 31−2s )L +1 L −35 + L −3 L +105 + (1 + 2 × 3−s + 31−2s )L +5 L −7 + L −15 L +21 ] L +1 L −403 + L +13 L −31 L +1 L −427 + L −7 L +61 2−1 [L +1 L −435 + L −3 L +145 + L +5 L −87 + L −15 L +29 ] 2−1 [L +1 L −483 + L −3 L +161 + L −7 L +69 + L −23 L +31 ] 2−1 [L +1 L −555 + L −3 L +185 + L +5 L −111 + L −15 L +37 ] 2−1 [L +1 L −595 + L +5 L −119 + L −7 L +85 + L +17 L −35 ] 2−1 [L +1 L −627 + L −3 L +209 + L −11 L +57 + L −19 L +33 ] 2−1 [L +1 L −715 + L +5 L −143 + L −11 L +65 + L +13 L −55 ] 2−1 [L +1 L −795 + L −3 L +265 + L +5 L −159 + L −15 L +53 ] 2−2 [L +1 L −1155 + L −3 L +385 + L +5 L −231 + L −7 L +165 + L −15 L +77 + L +21 L −55 + L +33 L −35 + L −11 L +105 ] 2−1 [L +1 L −1435 + L +5 L −287 + L −7 L +205 + L −35 L +41 ] 2−2 [L +1 L −1995 + L −3 L +665 + L +5 L −399 + L −7 L +285 + L −19 L +105 + L −15 L +133 + L +21 L −95 + L −35 L +57 ] 2−2 [L +1 L −3003 + L −3 L +1001 + L −7 L +429 + L −11 L +273 + L +13 L −231 + L +21 L −143 + L +33 L −91 + L −39 L +77 ] 2−2 [L +1 L −3315 + L −3 L +1015 + L +5 L −663 + L +13 L −255 + L +17 L −195 + L −15 L +221 + L −39 L +85 + L −51 L +85 ]

101 107 109 121 139 149 157 179 199 289 359 499 751 829

1.5 Contour integral technique

63

1.5 Contour integral technique We have seen that many lattice sums can be expressed as hyperbolic double sums. It appears that for several of these there exist representations as contour integrals. In at least one case the integral can be evaluated by residues and the lattice sum obtained in closed form, as we shall now illustrate. Consider the integral # π z 2l dz 1 (l = 0, 1, 2, . . .), Sl (n) = 2πi C (z 2 + n 2 )1/2 sinh π(z 2 + n 2 )1/2 sin π z (1.5.1) where C is the simple closed curve shown in Fig. 1.4. Note that the integrand has simple poles at z = ± p, ±i( p 2 +n 2 )1/2 , the where p = 0, 1, 2, . . .. By evaluating the integral with respect to the residues at poles lying within C, we obtain Sl (n) =

1 (−1)l n 2l−1 δl,0 − . 2n sinh (π n) 2 sinh (π n)

(1.5.2)

However, if we view C as a closed curve surrounding the remaining poles, we find Sl (n) = (−1)l

∞ 

(−1) p ( p 2 + n 2 )(l−1/2) cosech π( p 2 + n 2 )1/2

p=1

−

∞  p=1

(−1) p p 2l cosech π( p 2 + n 2 )1/2 . ( p 2 + ln 2 )1/2

(1.5.3)

On combining (1.5.2) and (1.5.3) we have ∞  p=1

(−1) p

[ p 2l + (−1)l+1 ( p 2 + n 2 )l ] 1 (−1)l n 2l−1 (1 − δl,0 ). (1.5.4) = 2 sinh (π n) ( p 2 + n 2 )1/2 sinh π( p 2 + n 2 )1/2

i √ n2 + 1 i|n| C

−2

−1

1

2

−i|n| −i √n2 + 1

Figure 1.4 Contour for the integration in Equation (1.5.1).

64

Lattice sums

Now, if F(n) is any function whatsoever, we have the identity ∞  ∞ 

(−1) p F(n)

[ p 2l + (−1)l+1 ( p 2 + n 2 )l ] ( p 2 + n 2 )1/2 sinh π( p 2 + n 2 )1/2

=

 n 2l−1 F(n) 1 (−1)l (1 − δl,0 ) . 2 sinh (π n)

n=1 p=1

∞

(1.5.5)

n=1

It is clear that a number of identities of this sort may be derived in a similar fashion, but we have not yet explored this to any great extent. When the function F(n) is a polynomial, the sums which occur on the righthand side of (1.5.5) were studied by Zucker [146]. Thus, if we take l = 1, F(n) = (−1)n+1 , we obtain ∞  ∞ 

(−1) p+n

n=1 p=1

(2 p 2 + n 2 ) 1 . = 8π ( p 2 + n 2 )1/2 sinh π( p 2 + n 2 )1/2

(1.5.6)

By interchanging p and n and adding the resulting sum to (1.5.6) there results ∞  ∞ 

(−1) p+n

n=1 p=1

( p 2 + n 2 )1/2 1 . = 2 2 1/2 12π sinh π( p + n )

(1.5.7)

In the calculation of the electric field gradient at an ion site in a body-centred cubic lattice one requires the sum  P2 (cos θ ) σ = , (1.5.8) R3 where θ is the angle between the lattice vector R and the z-axis and the sum is over the sublattice composing the structure that does not contain the site in question. This series is conditionally convergent, but by planewise summation de Wette [37] has shown that it can be put into the form σ = 2π 2



(−1) p+n

( p 2 + n 2 )1/2 , sinh π( p 2 + n 2 )1/2

(1.5.9)

where the sum is over all values of ( p, n) = (0, 0). This is easily expressed in terms of (1.5.7) and the sum ∞  (−1)n n n=1

sinh π n

=−

1 , 4π

(1.5.10)

whence we have σ =−

4π = −4.188790205 . . . , 3

(1.5.11)

whereas de Wette’s numerical result is σ = −4.189. It appears to us that this procedure is capable of being greatly extended and may lead to the exact evaluation

1.6 Conclusion

65

of many other useful lattice sums. A completely different class of contour integral representation was obtained by Glasser [56] using the Bromwich integral form for the inverse Mellin transform applied to the theta function representations discussed in Section 1.3. It will be interesting to see whether these integrals can be applied directly to the problem.

1.6 Conclusion The topic of lattice sums has roots and branches in widely differing areas of chemistry, physics, and mathematics. From a historical perspective, the subject appears as one having few themes but many variations. Of these we have attempted only to ring the changes on those themes which seem to us to be the most promising for future development and application. Thus we have discussed in some detail: (a) the θ -function approach, which subsumes the usual Ewald method; (b) the number-theoretic approach, which leads to many interesting and beautiful results whose utility is yet to be fully exploited. We have also briefly outlined the use of contour integration, which appears to offer a convenient and flexible representation of lattice sums and which has not yet been systematically explored. The following questions are deserving of further investigation. (1) Is it possible to develop alternative methods of systematically generating qseries which lie outside the realm of θ -functions? One of us (Glasser [59]) has found exactly a five-dimensional sum using q-series stemming from the Heine generalization of the hypergeometric function. (2) Is the conjecture (Glasser [57]) with regard to two-dimensional lattice sums correct? For, although we can only solve exactly for s = 1 those lattice sums to which the conjecture applies for all s, it is possible to find S1 (a, b, c), S2 (a, b, c), and S1,2 (a, b, c) exactly in terms of π and logarithms of surds for any a, b, and c (Zucker [145]). (3) Will the contour integral approach lead to simpler methods for calculating lattice sums? (4) Is there some connection between lattice sums and the well-known Watson integrals [138] for the corresponding lattice? We have not even touched on many-body lattice sums, which are of growing interest in chemistry and physics, nor on spatially restricted sums (Hoskins et al. [78]; Stepanets et al. [128]). We conclude with Samuel Johnson that ‘Nothing amuses more harmlessly than computation and nothing is more applicable to real business and speculative inquiry’.

66

Lattice sums

1.7 Appendix: Complete elliptic integrals in terms of gamma functions We wish here to outline the application of our results on two-dimensional sums to an old problem concerning complete elliptic integrals. Namely: when can K be expressed in terms of gamma functions for rational arguments? For this purpose, we consider the sums  2 (m + a 2 n 2 )−s , S0 (a, s) =  (−1)m (m 2 + a 2 n 2 )−s , S1 (a, s) =  (1.7.1) (−1)n (m 2 + a 2 n 2 )−s , S2 (a, s) =  (−1)m+n (m 2 + a 2 n 2 )−s , S3 (a, s) = over all pairs of integers (m, n) = (0, 0). It has been shown by two of the present authors (Glasser [58], Zucker [145]) that    2θ 3 (π α, q)   2  exp [2πi(nx + my)] 2π  1  2 y (1.7.2) ln  = 2π − , b 3|ab|  θ1 (0, q)  a 2 n 2 + b2 m 2 where q = exp (−π |a/b|), α = x + i y|a/b|, and thus   4k  2 π S1 (a, 1) = − ln , 3|a| k  2 4k π , ln S2 (a, 1) = − 3|a| k   4 π , ln S3 (a, 1) = − 3|a| kk 

(1.7.3)

where k 2 + k  2 = 1, K (k  ) = a K (k). Hence, when one of the Si (a, s) is solvable, k is expressible in terms of radicals. Furthermore, we have Kronecker’s Grenzformel (Selberg and Chowla [124])

    4|a| k  k 1/3 2π γ 2π π = − ln K . (1.7.4) lim S0 (a, s) − s→1 a(s − 1) |a| |a| π 4 Hence, when S0 and another Si are solvable, K can be expressed in closed form, as is illustrated by the following example. In the case a = 1, we have S0 S3

= 4β(s)ζ (s), = −4(1 − 21−s )ζ (s)β(s).

(1.7.5)

 From √ this we obtain S3 (1, 1) = −π ln 2 = −(π/3) ln (4/kk ). Therefore k = 1/ 2. From (1.7.4), we find

4   2( 34 ) π π = 2π γ + ln 2 − π ln 2 + π ln lim S0 − . (1.7.6) s→1 s−1 2 ( 14 )

1.8 Appendix: Watson integrals

67

√ √ From (1.7.6) we easily see that K (1/ 2) =  2 ( 14 )/4 π . For further details and a table of similar results, see Zucker [145].

1.8 Appendix: Watson integrals The generating functions for random walks on a lattice, along with a number of other ‘propagators’ for lattice phenomena, can be cast in the form  W =

dk . z − φ(k)

(1.8.1)

The integration in (1.8.1) is over the Brillouin zone and φ is a trigonometric polynomial in the components of k. Because G. N. Watson [138] was the first to evaluate an important family of these, they are now called Watson integrals. A great deal of effort has gone into evaluating these integrals for various lattices, much of which was summarized by Glasser and Zucker [64]; see also Hioe [76]. One particularly striking result is (Joyce [86], [87]) the following: I = = k± = ξ=

1 π3

 π π 0



0

π 0

du dv dw z − cos u − cos v − cos w

(ξ + 1)(ξ + 4) 8 2 2 π z ξ + 8ξ + 8 + 4(2 + ξ )(ξ + 1)1/2

1/2 K (k+ ) K (k− ),

(1.8.2)

ξ [(ξ + 1)1/2 ± (ξ + 4)1/2 ] − 2[1 − (ξ + 1)1/2 ] , ξ [(ξ + 1)1/2 ± (ξ + 4)1/2 ] + 2[1 − (ξ + 1)1/2 ] z 2 + 3 − [(z 2 − 9)(z 2 − 1)] ; z 2 − 3 + [(z 2 − 9)(z 2 − 1)]1/2

this result is for the simple cubic lattice. For the case z = 3 we find Watson’s formula √ √ √ √ √ √ √ 8 [10(17 − 12 2)]1/2 K [(2 − 3)( 3 + 2)] K [(2 − 3)( 3 − 2)] . 2 3π (1.8.3) By means of the sums for a 2 = 6 the method in Appendix 1.7 can be applied to evaluate the elliptic integrals in (1.8.3), and we obtain I =

√

I =

6 5 7 ( 1 )( 24 )( 24 )( 11 24 ). 96π 3 24

This raises the interesting question: is there a direct connection between lattice sums and Watson integrals for a given lattice?

68

Lattice sums

1.9 Commentary: Watson integrals In this commentary we sketch the method of determination of the classical Watson integrals and make some further elaborations on the previous appendix. More details on Watson integrals can be found in Chapter 9. 1.9.1 The easiest three-dimensional Watson integral We start with the easiest integral to evaluate. Let  π π π 1 d x d y dz, W3 (w) = 0 0 0 1 − w cos x cos y cos z for suitable w > 0. One may prove that  π π π 1 d x d y dz W3 (1) = 1 − cos x cos y cos z 0 0  0   = 14  4 14 = 4π K √1 , 2

via the binomial expansion and [24, Exercise 14, p. 188]. More generally (see below),

1 1 1   , 2, 2  2 3 2 2 2 ;4k 1 − k W3 ((2kk ) ) = π 3 F2 = 4π K 2 (k) . 1, 1 1.9.2 The harder three-dimensional Watson integrals We now describe results found largely in Joyce and Zucker [91, 92], where more background can also be found. The following integral arises in Gaussian and spherical models of ferromagnetism and in the theory of random walks. One of the most impressive closed-form evaluations of a multiple integral is that of Watson:  π π π 1 d x d y dz W1 = −π −π −π 3 − cos x − cos y − cos z     1 √ 1 = ( 3 − 1)  2 24  2 11 (1.9.1) 24 96  √ √ √  2 = 4π 18 + 12 2 − 10 3 − 7 6 K (k6 ) ,  √  √  √ 2− 3 3 − 2 is the sixth singular value; that is, √ K  (k6 )/K (k6 ) = 6. Elliptic integrals at the singular values may be evaluated in term of gamma functions; $ ∞ a large number of cases were already known in [54]. computation [91] Note that W1 = π 3 0 exp(−3t) I03 (t) dt allows for efficient $π here the Bessel function I0 (t) has been written as (1/π ) 0 exp[t cos θ ] dθ . The evaluation (1.9.1), in its original form, is due to Watson and is a tour de force. where k6

=

1.9 Commentary: Watson integrals

69

Next we describe a refined and simplified evaluation due to Joyce and Zucker [92]. Similarly, the integral  π π π d x d y dz W2 = 0 0 0 3 − cos x cos y − cos y cos z − cos z cos x   √ % & 2 1 = 3π K (sin( 12 (1.9.2) π )) = π4 2−2/3 β 2 13 , 13 ,   1 where sin 12 π = k3 is the third singular value. Indeed, as we shall see, (1.9.2) is easier and can be derived on the way to (1.9.1). The evaluation (1.9.2) then implies that  π π 1 dy dz W2 = 2 2 2 2 1/2 π 0 0 (9 − 8 cos y cos z − cos y − cos z + cos y cos z) after careful performance of the innermost integration. The expression inside the square root factors as (cos x cos y + cos x + cos y − 3)(cos x cos y − cos x − cos y − 3). Upon substituting s = x/2, and t = y/2, one obtains  π/2  π/2 dy dx % &% & 2 2 [ 1 − sin x sin y 1 − cos2 x cos2 y ]1/2 0 0 ∞ ∞  m + n  √   1  3 1 1 1 = 3 K 2 sin( 12 = β n + 2, m + 2 π) . 4π n m=0 n=0

1.9.3 More about the Watson integrals For a > 1, b > 1, one can show that 1 2  =

'  2(b+a) K (1+b)(1+a) dy = √ 1/2 (1 + b) (1 + a) 0 [(a + cos y) (b − cos y)] 1/2 π dt 

π

[(1 + b) (1 + a) cos2 t + (1 − a) (1 − b) sin2 t]1/2

0

.

A beautiful, but harder to establish, identity is (  (   π/2   1−c 1+c 2 2 2 K c cos s + sin s ds = K K , 2 2 0 (1.9.3) or equivalently 

π/2

K with

k

=

  % & 1 − (2kk  )2 cos2 θ dθ = K (k) K k 

0

√

1 − k 2 . Hence,   π/2 ' √ K 1 − (2k N k N )2 cos2 θ dθ = N K 2 (k N ) , 0

70

Lattice sums

where k N is the N th singular value. This is especially pretty for N = 1, 3, 7; 2 k N k N = 1, 12 , 18 , respectively. One may deduce that the face-centred cubic (FCC) lattice for the Green’s function evaluates as   π/2 ' √ 1 3 2 s + sin2 s ds = 3 K 2 (k ) . W2 = K cos 3 4 π 0 Correspondingly, Watson’s evaluation for the simple cubic (SC) lattice relied on deriving   π  √ cos x − 5 d x, W1 = 2 π K 2 0 and the following extension of (1.9.3):  π/2   K c2 cos2 s + d 2 sin2 s ds 0 ) )       1 − cd − (d 2 − 1)(c2 − 1) 1 + cd − (d 2 − 1)(c2 − 1) K . =K 2 2

1.9.4 The generalized Watson integrals Let W1 (w1 ) = W2 (w2 ) =

 π π π

d x d y dz

−π −π −π 3 − w1 (cos x − cos y − cos z)  π π π d x d y dz 0

0

,

0 3 − w2 (cos x cos y − cos y cos z − cos z cos x)

.

In a beautiful study, Joyce and Zucker [92], using the type of elliptic and hypergeometric transformations that we have explored, showed fairly directly that W2 (−w1 (3 + zw1 )/(1 − w1 )) = (1 − w1 )1/2 W1 (w1 ). It is now easy to verify that, with w1 = −1, this leads to a quite direct evaluation of (1.9.1) from (1.9.2). It is also true that  π/2   √ K 12 + 12 sin2 t dt. W1 = 2 π 0

One may also give a more symmetric form, namely   π/2 ' % & K 1 − 4k 2 1 − k 2 cos2 x d x  = 0

for 0 < k < 1.

0 π/2  π/2 0

dt d x [cos2 t

+ 4k 2 (1 − k 2 ) cos2

x sin2 t]1/2

1.9 Commentary: Watson integrals

71

Hint: For (1.9.3) consider N = 3 (c2 = 34 ) in (1.9.3), and let a and b be defined as a = (3 − cos x) / (1 + cos x) and b = (3 + cos x) / (1 − cos x) in (1.9.3). 1.9.5 Watson integral and Burg entropy Consider the perturbed Burg entropy maximization problem  1 1 1   v (α) := sup log ( p(x1 , x2 , x3 ))  p(x1 , x2 , x3 ) d x1 d x2 d x3 = 1 0 0 0 p≥0  1 1 1 and, for k = 1, 2, 3, p(x1 , x2 , x3 ) 0

0

0

 × cos (2π xk ) d x1 d x2 d x3 = α ,

maximizing the log of a density p with given mean and with the first three cosine moments fixed at a parameter value 0 ≤ α < 1. It transpires that there is a parameter value α such that below and at that value v(α) is attained while above that value it is finite $ but unattained. This is interesting, because the general method – maximizing T log ( p(t)) dt subject to a finite number of trigonometric moments – is frequently used. In one or two dimensions, such spectral problems are always attained when feasible. There is no easy way to see that this problem qualitatively changes at α, but we can get an idea by considering p (x1 , x2 , x3 ) =

1/W1 3 3 − 1 cos (2π xi )

and checking that this is feasible for α = 1 − 1/(3W1 ) ≈ 0.340537329550999142833; here α is expressed in terms of the first Watson integral, W1 . By using Fenchel duality [28] one can show that the above p is optimal. Indeed, for all α ≥ 0 the only possible optimal solution is of the form p α (x1 , x2 , x3 ) =

λ0α

−

3

1

i 1 λα

cos (2π xi )

,

for some real numbers λiα . Note that we have four coefficients to determine; using the four constraints we can solve for them. For 0 ≤ α ≤ α the precise form is parameterized by the generalized Watson integral: p α (x1 , x2 , x3 ) =

1/W1 (w) , 3 3 − 1 w cos (2π xi )

and α = 1 − 1/[3W1 (w)], from zero to one. Note also that the $ ∞ as3 w ranges 3 −3t expression W1 (w) = π 0 I0 (w t) e dt allows one to quickly obtain w from

72

Lattice sums

α numerically. For α > α, no feasible reciprocal polynomial can stay positive. Full details are given in [25].

1.10 Commentary: Nearest neighbour distance and the lattice constant The ‘size’ of the Madelung constant for any given crystal depends on whether one refers to the nearest neighbour distance or to the lattice constant. In the salt lattice, the nearest neighbour distance, that is, the distance between a sodium ion and a chlorine ion, is exactly half the distance between neighbouring sodium ions, which is the lattice constant. One will find both values scattered around in the literature. So, for NaCl, in terms of the nearest neighbour distance the Madelung constant is 1.74756. . . and in terms of the lattice constant it is 3.49513 . . . In Table 1.2 listing Madelung constants, all are given in terms of the lattice constant. In the early days of Madelung constants, when one had only very simple crystals with two or at most three ions to deal with, the nearest neighbour distance was the preferred choice, but when more complex crystals such as perovskite and spinel structures – which have large numbers of ions – were investigated, the lattice constant seemed a more sensible distance to refer to.

1.11 Commentary: Spanning tree Green’s functions On any graph G a spanning tree is a connected loop-free subgraph containing all the points (vertices) of G. It was proved by Temperley and others that if G is a regular lattice L restricted to N sites then the number of spanning trees TN (G) grows exponentially as N increases without bound and that the limit 1 log TN (G), N called the spanning tree constant for L, exists and is positive. Furthermore,  1 λL = d log[w − (θ )] d d θ, (1.11.1) π [0,π ]d λ L = lim

N →∞

where  is the structure function for L and w is the smallest positive (rational) value for which the integrand in (1.11.1) is singular. The spanning tree constant was first investigated for one- and two-dimensional lattices by Temperley [130]. Lin [99], Wu [140], Glasser and Wu [63], Chen and Wu [36], and Glasser and Lamb [62] evaluated the integral  2π  2π

1 log A + B + C − A cos θ1 − B cos θ2 − C cos(θ1 + θ2 ) dθ1 dθ2 , 2 4π 0 0 (1.11.2) which applies to the square and triangular lattices among others.

1.11 Commentary: Spanning tree Green’s functions

73

Higher-dimensional spanning tree constants were discussed by Shrock and Wu [127], who found, for a d-dimensional body-centred cubic lattice,    ∞ 1 1  1 (2l)! d (d) d log[1 − cos θ1 · · · cos θd ] d θ = , λ BCC = − d π [0,π ]d 2 l 22l l!2 l=1

and by Joyce [88], who independently showed, for d = 3, that  π π π 1 (3) log(1 − cos θ1 cos θ2 cos θ3 ) dθ1 dθ2 dθ3 λ BCC = − 3 π 0 0 0

3 3 3 , 2 , 2 , 1, 1 1 2 ;1 . (1.11.3) = 5 F4 16 2, 2, 2, 2 Here, we have used the generalized hypergeometric function, defined by    ∞ (a1 )n (a2 )n · · · (a p )n z n a1 , a2 , . . . , a p ;z = , p Fq b1 , b2 , . . . , bq (b1 )n (b2 )n · · · (bq )n n! n=0

where (a) p = (a + p)/ (a) is the Pochhammer symbol. Though it arises in a different context, we also mention the Baxter–Bazhanov formula (1993, unpublished)  1 log(2 − cos θ1 − cos θ2 − cos θ3 + cos θ1 cos θ2 cos θ3 ) d 3 θ λB B = 3 π [0,π ]3 8G = − 3 log 2, (1.11.4) π where G denotes Catalan’s constant. If we write  L = J(w) and treat w as a complex parameter then  dd θ 1 J  (w) = d (1.11.5) π [0,π ]d w − (θ) is the generalized Watson integral (Green’s function) for L, and thus the function J (w) is of considerable interest. Other than for d = 2, only a few closed-form evaluations are available, such as the Baxter–Bazhanov case studied by Joyce et al. [90]. However, this matter is closely tied in with the recent surge of interest in the logarithmic Mahler measure, defined by 

log P(e2πit1 , . . . , e2πitn ) d n t, (1.11.6) m(P) = [0,1]n

where P is an n-variate Laurent polynomial. For example, the value of m(4 − x − 1/x − y − 1/y), equivalent to a series identity due to Ramanujan, gives  π π 1 4G log(2 − cos θ1 − cos θ2 ) dθ1 dθ2 = λsq = 2 − log 2. π π 0 0

(1.11.7)

74

Lattice sums

Numerous conjectured and established values are available for a variety of polynomials, in the work of Boyd [30], Rogers [117], Bertin [13], and others. Several important three-dimensional cases have been announced recently. Thus a study of the Mahler measure by Joyce [89] relevant to the FCC lattice led to a formula for J (w) for the FCC case, and in the limit w → 3 he obtained λ FCC = 2λdiam  π π π 1 = 3 log(3 − cos θ1 cos θ2 − cos θ2 cos θ3 − cos θ3 cos θ1 ) dθ1 dθ2 dθ3 π 0 0 0

4 3 5 , , , 1, 1 14 8 8 3 2 3 = − log 2 + log 3 − ;1 . (1.11.8) 5 F4 15 5 135 2, 2, 2, 2 Guttmann and Rogers [66, 67] and Glasser [60] found J (w) for both the Baxter– Bahzanov case and the simple cubic case. For the simple cubic they obtain (by using modular parameterizations for hypergeometric functions) 

e−x − e−wx I03 (x) dx x 0

   R13 1 9 R13 1, 1, 54 , 32 , 74 2R0 = + ; 16 4 log 5 F4 5 8 R24 2, 2, 2, 2 (w 2 + 3)3 R2

 1, 1, 54 , 32 , 74 R1 3 R1 ; 16 4 + , (1.11.9) 5 F4 4 8 R3 2, 2, 2, 2 R3

JSC (w) =

∞

where √ R0 = w 3 − 5w − (w 2 − 1) w 2 − 9, √ R3 = w 2 − 9 − w w 2 − 9,

√ R1 = 2w 2 − 9 − 2w w 2 − 9, √ R4 = w 2 − 3 − w w 2 − 9.

A more direct connection between various Mahler measures and lattice sums is touched on in Chapter 6.

1.12 Commentary: Gamma function values in terms of elliptic integrals Since the first appearance of the work presented in this chapter the material of Appendix 1.7 has been considerably extended and the inverse question has been addressed, in [29]. This inverse question has complexity-theoretic implications, as it allows certain gamma and beta function values – such as π – to be computed as rapidly as complete elliptic integrals [23]. More recently, in 2008, Zucker and Broadhurst in previously unpublished computations provided the remarkable results in singular value theory given below.

1.12 Commentary: Gamma function values in terms of elliptic integrals 75 Let K (k) be the complete elliptic integral of the first kind, given by  1 dx  K (k) := , 2 0 (1 − x )(1 − k 2 x 2 )

(1.12.1)

where k denotes the modulus. For a given positive integer N , consider the equation K  (k) √ = N, (1.12.2) K (k) where K  (k) ≡ K (k  ) and k  ≡ (1 − k 2 )1/2 is the complementary modulus. When k ∈ (0, 1), it is known that (1.12.2) has a unique solution k ≡ k N which is an algebraic number – often, but not always soluble in radicals; these kn are the so-called singular values [24]. As also described in [24], Selberg and Chowla [123, 124] proved that K (k N ) ≡ K [N ] could be expressed in terms of products of  functions, algebraic numbers, and powers of π . Borwein and Zucker [29] showed that these formulae were much simplified if they were expressed in terms of the Euler beta function B( p, q), given by B( p, q) =

( p)(q) . ( p + q)

(1.12.3)

Further, for all N = 1, 2 (mod 4) an extra simplification was achieved, since it was found that only the central beta function, given by β( p) := B( p, p) =

 2 ( p) , (2 p)

(1.12.4)

was needed to express K [N ]. In every case the K [N ] become products of beta functions and algebraic numbers only. In the cases N = 1, 2 (mod 4) it was also found that β’s always appear in pairs β( p)β( 12 − p). So, making use of the relation β( 12 − p) = 24 p−1 tan(π p)β( p),

(1.12.5)

one could always reduce the argument of the β( p)’s involved so that p ≤ 14 . One really only needs to give K [N ] for N square-free. For, suppose that N = n 2 μ where n and μ are positive integers; then it is known that K [n 2 μ] = Mn (μ)K [μ],

(1.12.6)

where Mn (μ) is algebraic and is called the multiplier [24]. The following formulae for small Mn (μ) are known: M2 (μ) =

1 + kμ 2

27M34 (μ) − 18M32 (μ) − 8(1 − 2kμ2 )M3 (μ) − 1 = 0, (5M5 (μ) − 1) (1 − M5 (μ)) = 5

,

(1.12.7) (1.12.8)

256kμ2 (1 − kμ2 )M5 (μ). (1.12.9)

Many more multipliers are given in [24].

76

Lattice sums

These formulae for finding K [4N ], K [9N ], and K [25N ] from K [N ] depend only on knowing the singular value k N [24]. Thus, suppose we wish to find K [25] from K [1]. We substitute the value k12 = 1/2 into (1.12.9). In this particular √ case √ (1.12.9) can be solved, giving the result M5 = ( 5 + 2)/5, so K [25] = ( 5 + 2)K [1]/5. In compiling this compendium in appendix section A.3 at the back of the book, Zucker in 2005 first found (where possible) the values of kμ2 for 1 ≤ μ ≤ 25. Many of these results were obtained using the Mathematica routine ‘Recognize’. It will be seen that, for large N , associated with each β(n) is a factor tan(π n). Actually this is so for every N which is expressed in terms of central β functions, but for small N the tangents can be found in terms of radicals. In fact for 1 ≤ N ≤ 20, excluding N = 11 and 19, all K [N ] can be expressed in quadratic surds and at most three central beta functions. The most surprising of these is the case N = 17. Subsequently, in 2008, Broadhurst and Zucker extended the data in appendix section A.3 to include all K [N ] for 1 ≤ N ≤ 100, and this exercise required us to refine some views expressed above. (1) It became obvious that the best way to express the results most concisely √ was to use the function b( p) = β( p) tan( pπ ). (2) Because of a remarkable relation found empirically by David Broadhurst, it was possible to express every K [N ] in terms of the b( p) and algebraic quantities. It was well known that, for square-free N , the transcendental part of K [N ] contains the quantity G(D) :=

|D|−1  k=1

 

 D k (k ) , |D|

where % D & D = −4N if N ≡ 1, 2 (mod 4), D = −N if N ≡ 3 (mod 4), and k is the Legendre–Jacobi–Kronecker symbol. (3) Broadhurst discovered the beautiful general formula  2−( D) 2 (2π )h(D) G(D) = B(D)2 ,

B(D) =

(|D|−1/k)  k=1

 b

 D k (k ) , |D|

where h(D) is the class number of the binary quadratic form am 2 + bmn + cn 2 with fundamental discriminant D = (b%2 − & 4ac) < 0. For square-free N ≡ 1, 2 (mod 4), D = −4N is even and D2 = 0. For square-free N ≡ 3 (mod N ≡ 3 (mod 8) with & % D & 4), D = −N and two cases arise,% Dnamely 2 − 2 = 3 and N ≡ 7 (mod 8) with 2 − 2 = 1. Zucker’s method for finding K [N ] using disjoint and semi-disjoint discriminants was not available for most of the N values missing from the original list, but Broadhurst, using much more sophisticated numerical techniques, extracted all

1.13 Commentary: Integrals of elliptic integrals, and lattice sums

77

the required results. It seems that for N ≡ 2 (mod 4), K [N ]8 k 3N (1 − k 2N ) plays a pivotal role, with the invariants P[N ] :=

2k N , 1 − k 2N

used by Klein [24], or Q[N ] := P[N ]1/3 +

1 P[N ]1/3

being most easily found in radicals. For N ≡ 1 (mod 4), K [N ]8 k 2N (1 − k 2N ) appears as a feature, with either the quantity V [N ] := 4k 2N (1 − k 2N ) or the quantity W [N ] =: V [N ]1/3 +

1 V [N ]1/3

being more easily expressible in radicals. Also, for convenience, let T [N ] := k 3N (1 − k 2N ). Broadhurst has conjectured that the algebraic part of K [N ] can always be found in radicals despite the algebraic equations involved being of high degree. Indeed, equations of degree as high as 23 and 25 have been radicalized. For a list of K [N ], see appendix section A.3.

1.13 Commentary: Integrals of elliptic integrals, and lattice sums It is of some interest to consider integrals of the complete elliptic integrals E and K . Since the derivatives of E and K are again elliptic integrals, one may use integration by parts, for instance, to find some indefinite integrals, such as  x K (x) d x = E(x) + (x 2 − 1)K (x). A list of definite integrals involving K was compiled by Glasser [61], while a systematic study was that of Wan [136], where integrals of products such as K (x)2 are given in terms of hypergeometric functions. Evaluations of this kind, as well as neat identities like  1  1 K (x)2 d x = K  (x)2 d x, 2 0

0

78

Lattice sums

occur naturally in the study of uniform random walks in the plane (i.e., not on a lattice) and in Mahler measures; see [26, 27]. In [136], it was observed experimentally that    1  8 14 K  (x)3 d x = . (1.13.1) 128π 2 0 Closed-form evaluations of the integral of the cube function, such as $ 1 of a special 3 d x), are expected to be K (x) (1.13.1) and its many equivalent forms (e.g., 10 3 0 rare. A recent work by Rogers, Wan, and Zucker [118] gives a proof of (1.13.1). We start with the parametrizations (see [24]), with log q = −π K  (k)/K (k), K (k) =

π 2 θ (q), 2 3

k=

θ22 (q) θ32 (q)

k =

,

θ42 (q) θ32 (q)

,

k 4 dk = θ (q). dq 2q 4

Then we can write the integral in (1.13.1) as 

1

I =− 0

log3 q 2 θ (q)θ34 (q)θ44 (q) dq. 16q 2

Using a result of Ramanujan and η-function transformations, the integral can be written as  1 log3 q 4 η (q)η2 (q 2 )η4 (q 4 ) dq. I = −5 q 0 Now, with q = e2πiτ , f (τ ) = η4 (τ )η2 (2τ )η4 (4τ ) is a weight-5 modular form, which has the Fourier series f (τ ) =

∞ 

an e2πinτ ,

n=1

and therefore the last integral is the following L-series (as can be seen by taking a Mellin transform): I = 30

∞  an n=1

n4

.

(This is known as a central, or critical, L-value of f .) It can be shown by standard methods (see e.g., Section 5.1) that in fact f (τ ) =

∞ 1  2 2 (n − im)4 q n +m , 4 m,n=−∞

References

79

and so our integral reduces to

I =

15  (n − im)4 1 15  = , 2 2 4 2 m,n (n + m ) 2 m,n (n + im)4

√ which, from a computation in Section 4.8, is 2K (1/ 2)4 . Equation (1.13.1) now follows. The method outlined here can be used to establish connections between classes of integrals of E and K and lattice sums. Sometimes the integrals can be evaluated independently, therefore providing new results for lattice sums. Three examples are given here: 

(−1)m+1

m,n



(−1)m+n+1

m,n

 m,n

(−1)m+1

 2 ( 18 ) 2 ( 38 ) m 2 − 2n 2 , = 48π (m 2 + 2n 2 )2  6 ( 13 ) m 2 − 3n 2 = , 14 (m 2 + 3n 2 )2 2 3 π2  4 ( 14 ) m 2 − 4n 2 = . 32π (m 2 + 4n 2 )2

For more integrals involving K , see Chapter 9. We note that (1.13.1), among other integrals, has also been proved recently and independently in Zhou [141].

References [1] S. Adler. Generalized Ewald method for lattice sums. Physica (Utrecht), 27:1193– 1201, 1961. [2] P. Appell. Sur les fonctions de trois variables réelles satisfaisant à l’équation différentielle F = 0. Acta Math., 4:313–374, 1884. [3] P. Appell. Sur quelques applications de la fonction Z (x, y, z) à la physique mathématique. Acta Math., 8:265–294, 1886. [4] P. Appell. Sur les fonctions harmoniques à trois groupes de périodes. Rend. Circ. Mat. Palermo, 22:361–370, 1906. [5] M. N. Barber. Cross-over phenomena in the asymptotic behaviour of lattice sums. J. Phys. A., 10:2133–2142, 1977. [6] R. Bellman. A Brief Introduction to Theta Functions. Holt, New York, 1961. [7] G. C. Benson. A simple formula for evaluating the Madelung constant of a NaCltype crystal. Can. J. Phys., 34:888–890, 1956.

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[8] G. C. Benson and H. P. Schreiber. A method for the evaluation of some lattice sums occurring in calculations of physical properties of crystals. II. Can. J. Phys., 33:529–533, 1955. [9] G. C. Benson, H.P. Schreiber, and D. Patterson. An examination of Verwey’s model for the lattice structure of the free surface of alkali halide crystals. Can. J. Phys., 34:265–275, 1956. [10] G. C. Benson and F. van Zeggeren. Madelung constants of some cubic crystals. J. Chem. Phys., 26:1083–1085, 1957. [11] F. Bertaut. L’énergie électrostatique de réseaux ioniques. J. Phys. Radium, 13:499– 505, 1952. [12] F. Bertaut. C. R. Hebd. Seances Acad. Sci., 239:234–235, 1954. [13] M. J. Bertin. Mesure de Mahler d’hypersurfaces K 3. J. Number Theory, 128:2890– 2913, 2008. [14] H. Bethe. Splitting of terms in crystals. Ann. Phys. (Leipzig), 3:137, 1929. [15] J. L. Birman. Effect of overlap on electrostatic lattice potentials in ionic crystals. Phys. Rev., 97:897–902, 1955. [16] M. Boon and J. Zak. Coherent states and lattice sums. J. Math. Phys., 19:2308– 2311, 1978. [17] M. Born. Dynamik der Kristallgitter. Teubner, Leipzig, 1916. [18] M. Born. Über elektrostatische Gitterpotentiale. Zeitschrift für Physik, 7:124–140, 1921. [19] M. Born and M. Bradburn. The thermodynamics of crystal lattices. II. Proc. Camb. Phil. Soc., 39:104–113, 1943. [20] M. Born and R. Fürth. The stability of crystal lattices. III. Proc. Camb. Phil. Soc., 36:454–465, 1940. [21] M. Born and M. Göppert-Mayer. In Handbuch der Physik (S. Fliigge, ed.), vol. 24, Part 2, p. 707. Springer-Verlag, Berlin and New York, 1933. [22] M. Born and R. D. Misra. On the stability of crystal lattices. IV. Proc. Camb. Phil. Soc., 36:466–478, 1940. [23] J. M. Borwein and D. H. Bailey. Mathematics by Experiment: Plausible Reasoning in the 21st Century, 2nd edition. A. K. Peters, 2008. [24] J. M. Borwein and P. B. Borwein. Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity. Wiley, New York, 1987. [25] J. M. Borwein and A. S. Lewis. Partially-finite convex programming in L 1 : entropy maximization. SIAM J. Optimization, 3:248–267, 1993. [26] J. M. Borwein, A. Straub, and J. G. Wan. Three-step and four-step random walk integrals. Exp. Math., 22:1–14, 2013. [27] J. M. Borwein, A. Straub, J. G. Wan, and W. Zudilin. Densities of short uniform random walks. Can. J. Math., 64:961–990, 2012, with an appendix by Don Zagier. [28] J. M. Borwein and J. Vanderwerff. Convex Functions: Constructions, Characterizations and Counterexamples. Encyclopedia of Mathematics and Its Applications, vol. 109, Cambridge University Press, 2010. [29] J. M. Borwein and I. J. Zucker. Elliptic integral evaluation of the Gamma function at rational values of small denominator. IMA J. Numer. Anal., 12:519–526, 1992.

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[53] F. G. Fumi and M. P. Tosi. Extension of the Madelung method for the evaluation of lattice sums. Phys. Rev., 117:1466–1468, 1960. [54] J. W. L. Glaisher. Messenger of Mathematics, vol. 24, p. 27, 1895. [55] M. L. Glasser. A Watson sum for a cubic lattice. J. Math. Phys., 13:1145, 1972. [56] M. L. Glasser. The evaluation of lattice sums. I. Analytic procedures. J. Math. Phys., 14:409–413, 1973a. [57] M. L. Glasser. The evaluation of lattice sums. II. Number-theoretic approach. J. Math. Phys., 14:701–703, 1973b. [58] M. L. Glasser. The evaluation of lattice sums. III. Phase modulated sums. J. Math. Phys., 15:188–189, 1974. [59] M. L. Glasser. The evaluation of lattice sums. IV. A five-dimensional sum. J. Math. Phys., 16:1237–1238, 1975. [60] M. L. Glasser. A note on a hyper-cubic Mahler measure and associated Bessel integral. J. Phys. A, 45:494002, 2012. [61] M. L. Glasser. Definite integrals of the complete elliptic integral K . J. Res. NBS, 80B:313–323, 1976. [62] M. L. Glasser and G. Lamb. A lattice spanning tree entropy function. J. Phys. A: Math. Gen., 38:L471–L475, 2005. [63] M. L. Glasser and F. Y. Wu. On the entropy of spanning trees on a large triangular lattice. Ramanujan Journal, 10:205–214, 2005. [64] M. L. Glasser and I. J. Zucker. Extended Watson integrals for the cubic lattices. Proc. Nat. Acad. Sci. USA, 74:1800–1801, 1977. [65] A. Graovac, H. J. Monkhorst, and M. L. Glasser. Computation of Fourier transform quantities in Hartree–Fock calculations for simple crystals. Int. J. Quantum Chem., 9:243–259, 1975. [66] A. J. Guttmann and M. Rogers. An integral arising from the chiral sl(n) Potts model. J. Phys. A: Math Theor., 46:045202, 2013. [67] A. J. Guttmann and M. Rogers. Spanning tree generating functions and Mahler measures. J. Phys. A., to appear. [68] G. E. Hall. Asymptotic properties of generalized Chaba and Pathria lattice sums. J. Math. Phys., 17:259–260, 1976a. [69] G. E. Hall. Weak phase transitions in asymptotic properties of lattice sums. J. Stat. Phys., 14:521–524, 1976b. [70] G. E. Hall. Order relations for lattice sums from order relations for theta functions. J. Phys. Chem. Solids, 38:367–373, 1977. [71] G. H. Hardy. On some definite integral considered by Mellin. Messenger of Mathematics, vol. 49, pp. 85–91, 1919. [72] F. E. Harris and H. J. Monkhorst. Electronic-structure studies of solids. I. Fourier representation method for Madelung sums. Phys. Rev. B, 2:4400–4405, 1970. [73] A. Hautot. A new method for the evaluation of slowly convergent series. J. Math. Phys., 15:1722–1727, 1974. [74] A. Hautot. New applications of Poisson’s summation formula. J. Phys. A, 8:853– 862, 1975. [75] W. R. Heller and A. Marcus. A note on the propagation of excitation in an idealized crystal. Phys. Rev., 84:809–813, 1951.

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[145] I. J. Zucker. The evaluation in terms of -functions of the periods of elliptic curves admitting complex multiplication. Math. Proc. Camb. Phil. Soc., 82:111–118, 1977. [146] I. J. Zucker. The summation of series of hyperbolic functions. SIAM J. Math. Anal., 10:192–206, 1979. [147] I. J. Zucker and M. M. Robertson. Some properties of Dirichlet L-series. J. Phys. A., 9:1207–1214, 1976. [148] I. J. Zucker and M. M. Robertson. Systematic approach to the evaluation of  2 2 −s (m,n=0,0) (am + bmn + cn ) . J. Phys. A, 9:1215–1225, 1976.

2 Convergence of lattice sums and Madelung’s constant

The lattice sums involved in the definition of Madelung’s constant for an NaCl–type crystal lattice in two or three dimensions are investigated here. The fundamental mathematical questions of convergence and uniqueness of the sum of these series, which are not absolutely convergent, are considered. It is shown that some of the simplest direct sum methods converge and some do not converge. In particular, the very common method of expressing Madelung’s constant by a series obtained from expanding spheres does not converge. The concept of the analytic continuation of a complex function to provide a basis for an unambiguous mathematical definition of Madelung’s constant is introduced. By these means, the simple intuitive direct sum methods and the powerful integral transformation methods, which are based on theta function identities and the Mellin transform, are brought together. A brief analysis of a hexagonal lattice is also given.

2.1 Introduction Lattice sums have played a role in physics for many years and have received a great deal of attention on both practical and abstract levels. The term ‘lattice sum’ is not a precisely defined concept: it refers generally to the addition of the elements of an infinite set of real numbers, which are indexed by the points of some lattice in N -dimensional space. A method of performing a lattice sum involves accumulating the contributions of all these elements in some sequential order. Unfortunately, the elements of the set are not, in general, absolutely summable so the sequential order chosen can affect the answer. In this chapter we are concerned with the particular lattice sums involved in the Madelung constant. Indeed, attaining specificity in the definition of the Madelung constant is our primary motive. Although we are dealing with purely mathematical questions, it is our belief that

88

Convergence of lattice sums and Madelung’s constant

the results presented here may shed some light on the physics of crystals. Other researchers [7, 8, 20] have expressed concern about the ambiguities involved in summing a non-absolutely-convergent series in a different manner, but it appears that no one has confronted this question fully. Let L be a lattice in N -dimensional space and let A L = {al : l ∈ L} be a set of real numbers indexed by L. There are two basic approaches to summing the elements of A L : by direct summation or by integral transformations. The major factors involved in choosing a method are physical meaningfulness and speed of convergence. The direct summation methods involve an orderly grouping of the elements of A L into sequentially indexed finite subsets increasing in size to eventually include any element of A L . Sometimes fractions of elements are included in the subsets to maintain a physical principle such as electrical neutrality. Two commonly used direct summation methods are due to Evjen [16] and Højendahl [23]. The most commonly used integral transformation method is known as the Ewald method [17]. More recently the Mellin transformation applied to theta functions has been used to put the integral transformation methods in a general context. An excellent review article by Glasser and Zucker [20] gives a development of these methods and an extensive bibliography. In this chapter we deal primarily with NaCl-type ionic crystals in two or three dimensions. This is for two main reasons, the ease of notation in these cases and the fact that almost every textbook introduces the Madelung constant on this type of crystal first. From a mathematical and physical point of view there are two very reasonable and simple direct summation methods that could be applied to give the Madelung constant for an NaCl-type ionic crystal. One could take a basic cube centred at the referenced ion with sides parallel to the basic vectors and let the cube expand as the contributions from all lattice points within the cube are accumulated. Alternately one could use expanding spheres centred at the reference ion. This latter method is intuitively appealing since all ions that are an equal distance from the reference ion are given equal treatment. Thus, many textbooks [e.g., 1, 26] some research articles) write down the resulting infinite √ √ (and √ series (6 − 12/ 2 + 8/ 3 − 6/ 4 + · · · ) as giving the Madelung constant for an NaCl-type ionic crystal. Unfortunately, this infinite series does not converge. This was proven by Emersleben [15] and, in light of the fact that most people are unaware of this divergence, we include a short elementary proof in Theorem 2.3. Section 2.2 is devoted to the two-dimensional square lattice while Section 2.3 contains the above-mentioned result on expanding spheres. In Theorem 2.4 we prove that the method of expanding cubes converges. In Section 2.4, the mathematical tools become more sophisticated as we consider integral transformation methods and their relation to the direct summation methods dealt with in Section 2.3. We have included, in Section 2.5, a careful analysis of some direct summation methods in two dimensions in light of their property of being analytic in the inverse power exponent. This analysis is quite illustrative of the relations

2.2 Two dimensions

89

between the various summation methods. In Section 2.6 we do a brief analysis of a two-dimensional hexagonal lattice. Section 2.7 gives our conclusions.

2.2 Two dimensions It is convenient to introduce the notation in the two-dimensional case of a simple lattice in the plane with unit charges located at integer lattice points ( j, k) and of sign (−1) j+k . The potential energy at the origin due to the charge at ( j, k) is −(−1) j+k /( j 2 + k 2 )1/2 . If we want the total potential energy at the origin due to all other charges, then we must sum all the numbers in the following set: + * A = (−1) j+k /( j 2 + k 2 )1/2 : ( j, k) ∈ Z/(0, 0) , where Z denotes the integers. Because the elements of the subset of A with j = k form a set of positive numbers with divergent sum, it is clear that the value of the sum is highly dependent on the order in which the elements of A are added. It is not immediately clear that any reasonable method will produce a convergent series. In addition, for the model to be physically relevant, all ‘reasonable’ methods should converge to the same number. Here are two very reasonable methods. First, consider the total potential due to all the charges within a circle of radius r about the origin and let r → ∞. This leads to the series ∞  (−1)n C2 (n)

n 1/2

n=1

,

(2.2.1)

where C2 (n) is the number of ways of writing n as the sum of two squares of integers (positive, negative, or zero). In deriving (2.2.1), we use the fact that 2 2 (−1) j+k = (−1) j +k = (−1)n , for any j, k ∈ Z with j 2 + k 2 = n. We will refer to (2.2.1) as the method of expanding circles. Second, there is the method of expanding squares. This is intuitively appealing, as a perfect crystal grows by expansion of the shape of a basic unit cell. For each natural number n, let    (−1) j+k : −n ≤ j, k ≤ n and ( j, k)  = (0, 0) . (2.2.2) S2 (n) = ( j 2 + k 2 )1/2 Then lim S2 (n)

n→∞

is a way of expressing the series obtained by expanding squares. It turns out that both these methods converge, as we will now show. Theorem 2.1

The series in (2.2.1) converges.

Proof To carry out the proof that the series in (2.2.1) converges we introduce some notation and standard facts from number theory. For any sequence of real

90

Convergence of lattice sums and Madelung’s constant

numbers {an }∞ β real we write an = O(n β ) if the sequence {n −β an } is n=1 and n bounded. Let An = k=1 C2 (k), for each natural number n. Then An denotes the number of non-origin lattice points inside or on a circle of radius n 1/2 . It is fairly easy to see that An should be approximately π n; in fact, the reader can easily show that An − π n = O(n 1/2 ). However, this is not quite good enough for us here so we quote a stronger result, which can be found in Dickson [14]: An − π n = O(n α )

for some α,

1 4

< α < 13 .

(2.2.3)

For a natural number n, a divisor of n is a natural number d such that d divides n. Let d(n) denote the number of divisors of n and let dk (n) denote the number of divisors d of n with d congruent to k modulo 4, for k = 1 or 3. With this notation, Theorem 278 of Hardy and Wright [22] implies that C2 (n) = 4[d1 (n) − d3 (n)].

(2.2.4)

This together with Theorem 315 of Hardy and Wright [22] gives C2 (n) = O(n δ )

for any δ > 0.

(2.2.5)

Note that dk (2n) = dk (n) for k = 1 or 3 and any n. So C2 (2n) = C2 (n) Let Bn =

n

k=1 (−1)

kC

2 (k).

B2n =

for all natural numbers n.

(2.2.6)

Then 2n  (−1)k C2 (k) k=1

=

n 

C2 (2k) −

k=1

=2

n 

C2 (2k − 1)

k=1

n  k=1

C2 (2k) −

2n 

C2 (k).

k=1

Using (2.2.6), B2n = 2An − A2n . From (2.2.3) and (2.2.7), we have that, with α as in (2.2.3), B2n = O(n α ). Furthermore, this along with (2.2.5) implies that B2n+1 = B2n − C2 (2n + 1) = O(n α ).

(2.2.7)

2.2 Two dimensions

91

Therefore Bn = O(n α ).

(2.2.8)

Now consider the partial sums of the series in (2.2.1): Tn =

n  (−1)k C2 (k)

k 1/2

k=1

= O(n α−1/2 ) −



Bn = + Bk k −1/2 − (k + 1)−1/2 (n + 1)1/2 n

k=1

n 



Bk (k + 1)−1/2 − k −1/2 .

(2.2.9)

k=1

By the mean value theorem, [(k + 1)−1/2 − k −1/2 ] ≤ 12 k −3/2 and therefore 

  Bk (k + 1)−1/2 − k −1/2  = O(k α−3/2 ).  [(k + 1)−1/2 − k −1/2 ] converges absolutely. Since Since α − 32 < −1, ∞ k=1 Bk  1 −1/2 − k −1/2 ] exists. That is, the α − 2 < 0, limn→∞ Tn = − ∞ k=1 Bk [(k + 1) series in (2.2.1) converges. We now turn to the limit in (2.2.2). We need an easy lemma from calculus that will be left to the reader to verify. This lemma will also be used in the proof of Theorem 2.4. Lemma 2.1 For any positive real numbers a, b, and s, each of the following functions is monotonically decreasing in t, for 0 < t < ∞: f 1,s (t) = t −s , f 2,s (t) = t −s − (t + a)−s , f 3,s (t) = t −s − (t + a)−s − (t + b)−s + (t + a + b)−s . Theorem 2.2

The limit in (2.2.2) exists.

Proof We apply the lemma to f 2,s with a = (k + 1)2 − k 2 and s = 12 . Then, if j ≥ 0 and k ≥ 0 with j + k ≥ 1, we have f 2,1/2 ( j 2 + k 2 ) > f 2,1/2 (( j + 1)2 + k 2 ). Explicitly, we have ( j 2 + k 2 )−1/2 − [ j 2 + (k + 1)2 ]−1/2 − [( j + 1)2 + k 2 ]−1/2 + [( j + 1)2 + (k + 1)2 ]−1/2 > 0.

(2.2.10)

Let g( j, k) denote the left-hand side of (2.2.10). Then (−1) j+k g( j, k) is the contribution to the potential at the origin due to a basic cell of four adjacent ions with the closest ion at ( j, k). Inequality (2.2.10) says that the contribution always has the same sign as that of the nearest ion.

92

Convergence of lattice sums and Madelung’s constant Rewrite S2 (n) using the symmetries to get S2 (n) = 4Q(n) + 4X (n),

where Q(n) = X (n) =

n  j,k=1 n  k=1

(−1) j+k , ( j 2 + k 2 )1/2

(−1)k . k

Since limn→∞ X (n) = − ln 2, if we prove that limn→∞ Q(n) exists then the limit in (2.2.2) will exist. We will establish a number of properties of the sequence {Q(n)}∞ n=1 , which will be used to prove its convergence. Property 1: Q(2n) − Q(2n − 2) > 0

for all n ≥ 2.

That is, the even indexed elements increase. To see this we group the terms of Q(2n) − Q(2n − 2) into basic cells of four, as illustrated in Fig. 2.1a for Q(6) − Q(4). Thus, Q(2n) − Q(2n − 2) =

n 

g(2l − 1, 2n − 1) +

l=1

n−1 

g(2n − 1, 2m − 1),

m=1

where as before, g( j, k) denotes the left-hand side of (2.2.10). So property 1 holds. Property 2: Q(2n + 1) − Q(2n − 1) < 0 for all n ≥ 1. 7

7

6

6

5

5

4

4

3

3

2

2

1

1 1

2

3

4 (a)

5

6

7

1

2

3

4

5

(b)

Figure 2.1 Illustrations of (a) property 1 and (b) property 2.

6

7

2.3 Three dimensions

93

That is, the odd-indexed elements of the sequence decrease. Referring to Fig. 2.1b and correcting for the overlap at the (2n, 2n) point we are led to the following grouping: Q(2n + 1) − Q(2n − 1) =

n n   [−g(2l − 1, 2n)] + [−g(2n, 2m − 1)] l=1



m=1

 1 1 − √ − √ < 0. n 2 (n + 1) 2 Property 3: Q(2n + 1) − Q(2n) > 0 for all n ≥ 1. Thus, the odd-indexed elements are all greater than any even-indexed element. This is clear again from a simple grouping of terms and using the monotonicity of f 1,1/2 from the lemma: n   1 Q(2n + 1) − Q(2n) = 2 ((2l − 1)2 + (2n + 1)2 )1/2 l=1  1 1 + − √ > 0. (4l 2 + (2n + 1)2 )1/2 (2n + 1) 2 Property 4: lim Q(2n + 1) − Q(2n) = 0.

n→∞

Thus, the difference between successive elements goes to zero. To see this, simply note that 1 2 + 0 < Q(2n +1)− Q(2n) < √ → 0 as n → ∞. [1 + (2n + 1)2 ]1/2 (2n + 1) 2 It is now easy to see that properties 1–4 imply that limn→∞ Q(n) exists. This completes the proof of Theorem 2.2. Thus, we have shown that two of the most obvious methods of summing for a Madelung constant in two dimensions each converge. At this point, no indication has been given that the two methods yield the same number. That this is indeed so will be shown in Section 2.5.

2.3 Three dimensions In this section the three-dimensional case will be considered. For the Madelung constant of an NaCl-type crystal one must investigate ways of summing the elements of the following set: + * B = (−1) j+k+l /( j 2 + k 2 + l 2 )1/2 : ( j, k, l) ∈ Z3 /(0, 0, 0) .

94

Convergence of lattice sums and Madelung’s constant

In analogy with the two-dimensional case we will consider the method of expanding spheres about the origin and the method of expanding cubes. Our next theorem is a negative result, which is quite startling. Many textbooks in physical chemistry and solid state physics give the series dealt with in Theorem 2.3 as the Madelung constant for an NaCl-type crystal [1, 26]. It also appears in research articles and undergraduate courses. Although no one sums this series directly, it is physically misleading to believe that it converges to anything. Let C3 (n) denote the number of ways of writing n as a sum of three squares. If we consider a sphere centred at the origin in three-space, add all the elements of B that correspond to lattice points within the sphere, and then let the radius go to infinity, we are led to the infinite series ∞  (−1)n C3 (n) . √ n

(2.3.1)

n=1

Theorem 2.3 (Emersleben [15]) The series in (2.3.1) diverges. Proof It is interesting that the proof that the series in (2.3.1) diverges is much less sophisticated than the proof in Theorem 2.1 that the series in (2.2.1) converges. Our main tool is a simple estimate of the number of non-zero lattice points on √ or √ inside a sphere of radius r . Call this number L r . Notice that, for n ≤ r < n + 1, n  C3 (k). Lr = k=1

We leave to the reader the easy task of verifying that 4 L r − πr 3 = O(r 2 ). 3 This implies that Lr 4 = π. 3 r3 Proceeding with a proof by contradiction we assume that lim

r →∞

(2.3.2)

∞  (−1)n C3 (n) converges. √ n n=1 √ This implies that n = C3 (n)/ n → 0 as n → ∞. For a natural number N , let M N = max{n : n ≥ N }. Then M N → 0 as N → ∞. Fix N for the moment and consider, for n > N , the inequality        n N n √ √ √ L √n k k ≤ n −3/2 k k + M N n −3/2 k . √ 3 = n −3/2 ( n) k=1 k=1 k=N +1 (2.3.3)

2.3 Three dimensions Now

 n  √ k≤ k=N +1

n+1 N +1

t 1/2 dt =

2 3

95



(n + 1)3/2 − (N + 1)3/2 .

Inserting this in (2.3.3) implies that        N √ L √n n + 1 3/2 N + 1 3/2 −3/2 2 . ≤n k k + 3 M N − √ n n ( n)3 k=1

√ Letting n → ∞, we see that lim supn→∞ L √n /( n)3 ≤ 23 M N , for any N . Since M N → 0 as N → ∞, we have that L √n √ 3 = 0. n→∞ ( n) lim

This is a contradiction of (2.3.2). Therefore, ∞  (−1)n C3 (n) √ n

diverges.

n=1

In fact, the contributions of individual spherical shells do not tend to zero. So it is not at all appropriate to define the Madelung constant via the method of expanding spheres. We turn to the method of expanding cubes. Let    (−1) j+k+l : −n ≤ j, k, l ≤ n, ( j, k, l) = (0, 0, 0) . S3 (n) = ( j 2 + k 2 + l 2 )1/2 Theorem 2.4

The limit limn→∞ S3 (n) exists.

Proof We proceed as in the proof of Theorem 2.2. For j, k, l ≥ 1 let g( j, k, l) = ( j 2 + k 2 + l 2 )−1/2 − [( j + 1)2 + (k + 1)2 + (l + 1)2 ]−1/2 + [( j + 1)2 + (k + 1)2 + l 2 )−1/2 + [( j + 1)2 + k 2 + (l + 1)2 ]−1/2 + [ j 2 + (k + 1)2 + (l + 1)2 ]−1/2 − [( j + 1)2 + k 2 + l 2 ]−1/2 − [ j 2 + (k + 1)2 + l 2 ]−1/2 − [ j 2 + k 2 + (l + 1)2 ]−1/2 . Then (−1) j+k+l g( j, k, l) represents the contribution to the potential at the origin of a basic unit cell whose closest corner is at ( j, k, l). An appropriate use of the monotonicity of f 3,1/2 from Lemma 2.1 shows that g( j, k, l) > 0 for all j, k, l ≥ 1.

96

Convergence of lattice sums and Madelung’s constant

Let h(k, l) = (2k 2 + l 2 )−1/2 − [2k 2 + (l + 1)2 ]−1/2 − [2(k + 1)2 + l 2 ]−1/2 + [2(k + 1)2 + (l + 1)2 ]−1/2 . Using f 2,1/2 , we get that h(k, l) > 0 for all k, l ≥ 1. Let P(n) denote the part of S3 (n) that comes from the positive octant. That is, for n ≥ 1, n  (−1) j+k+l . P(n) = ( j 2 + k 2 + l 2 )1/2 j,k,l=1

Then, in a manner similar to that used for the Q(n), it can be shown that limn→∞ P(n) exists. We proceed with the details of this demonstration. The following identities are most easily seen by drawing a three-dimensional version of Fig. 2.1, but they can be verified directly: P(2n + 1) − P(2n − 1) = 3

n n   % & g 2n, 2k − 1, 2l − 1 + 3 h(2n, 2 j − 1) k,l=1

j=1

  1 1 1 +√ − , 2n + 1 3 2n n  P(2n + 2) − P(2n) = −3 g(2n + 1, 2k − 1, 2l − 1) −3

(2.3.4)

k,l=1 n 

g(2n + 1, 2n + 1, 2 j − 1)

j=1

− g(2n + 1, 2n + 1, 2n + 1).

(2.3.5)

Both (2.3.4) and (2.3.5) hold for all n ≥ 1. From (2.3.4) and (2.3.5) we get the properties of odd- or even-element monotonicity of the sequence of P(n). Property 1 : P(2n) − P(2n − 2) < 0 for all n ≥ 2. Property 2 : P(2n + 1) − P(2n − 1) > 0

for all n ≥ 1.

Notice that the inequalities are reversed from those of properties 1 and 2 in the two-dimensional case. To get the analogues of properties 3 and 4 for the P(n) we need to refer to the lemma for a final time. For n, j, k ≥ 1, let h 0 (n, j, k) = (n 2 + j 2 + k 2 )−1/2 − [n 2 + ( j + 1)2 + k 2 ]−1/2 − [n 2 + j 2 + (k + 1)2 ]−1/2 + [n 2 + ( j + 1)2 + (k + 1)2 ]−1/2 .

2.3 Three dimensions

97

With a = (k + 1)2 − k 2 , h 0 (n, j, k) = f 2,1/2 (n 2 + j 2 + k 2 ) − f 2,1/2 (n 2 + ( j + 1)2 + k 2 ), which is positive for all n, j, k ≥ 1. With this notation, P(2n + 1) − P(2n) = −3 −3

n 

h 0 (2n + 1, 2 j − 1, 2k − 1)

j,k=1 n 

*

[2(2n + 1)2 + (2l − 1)2 ]−1/2

l=1

− [2(2n + 1)2 + (2l)2 ]−1/2 −

1

√ . (2n + 1) 3

+ (2.3.6)

This leads to the following property. Property 3 : P(2n + 1) − P(2n) < 0 for all n ≥ 1. Therefore, the decreasing even-indexed elements are all greater than the increasing odd-indexed elements. To see that there is a unique limit to the sequence of P(n), we only need the last property, which implies that the distance between successive terms approaches zero. This follows from (2.3.6) and P(2n + 1) − P(2n) > −

3 1 − √ . 2 1/2 [(2n + 1) + 2] (2n + 1) 3

(2.3.7)

To verify (2.3.7) let xj =

2n  k=1

(−1)1+ j+k [(2n + 1)2 + j 2 + k 2 ]1/2

for 1 ≤ j ≤ 2n + 1.

Using the function h 0 , defined above, write |x j | − |x j+1 | =

n 

h 0 (2n + 1, j, 2k − 1) > 0.

(2.3.8)

k=1

Note that x j itself is an alternating sum of decreasing terms, so the sign of x j is (−1) j . With (2.3.8), this implies that 0>

2n+1  j=1

x j > x1 >

−1 . [(2n + 1)2 + 2]1/2

98

Convergence of lattice sums and Madelung’s constant

Then P(2n + 1) − P(2n) = 3

2n+1 

xj −

j=1

3 √ >− [(2n + 1)2 + 2]1/2 (2n + 1) 3 1

1 √ . (2n + 1) 3

−

So (2.3.7) holds. Thus, the following property has been established. Property 4 : lim P(2n + 1) − P(2n) = 0.

n→∞

Properties 1 –4 imply that limn→∞ P(n) exists. Finally, S3 (n) = 8P(n) + 12Q(n) + 6X (n),

(2.3.9)

where, as before, Q(n) =

n  j,k=1

(−1) j+k ( j 2 + k 2 )1/2

and X (n) =

n  (−1)k k=1

k

.

Since each term on the right-hand side of (2.3.9) approaches a limit as n → ∞, we have that limn→∞ S3 (n) exists. Remark 1: Although this method of summing over expanding cubes is not rapidly convergent, it is extremely well behaved. The alternations of P(n) and Q(n) above and below their limiting values provide precise error bounds, which may be useful in theoretical considerations. Remark 2: The work of Campbell [8] must be mentioned at this point. He states general conditions on a doubly indexed series and concludes a convergence result, which is stronger than Theorem 2.2 above. However, there is a serious error in his proof and his general theorem is false. A simplified version of Campbell’s claimed result would be the following. Let {ai j }i∞= 1 ,∞ j = 1 be a doubly indexed ‘sequence’ of reals satisfying: (i) for all i, {|ai1 |, |ai2 |, |ai3 |, . . .} is a monotonically decreasing sequence with lim j→∞ ai j = 0 and, for all j, {|a1 j |, |a2 j |, |a3 j |, . . .} is a monotonically decreasing sequence with limi→∞ ai j = 0; (ii) the sign of ai j is ∞ ∞ ( j=1 ai j ) exists. (−1)i+ j+1 . Then i=1 Here is a counterexample to this claim. Let the ai j be defined as in the array in Table 2.1. Let ∞  ai j for i = 1, 2, 3, . . . Ui = j=1

2.3 Three dimensions

99

Table 2.1 An array ai j that forms a counterexample to the general convergence result claimed by Campbell i\ j 1 2 3 4 5 6 . . .

1

2

1 − 12 − 10−2

− 21 1 − 10−3 2 − 31 1 − 10−4 3 − 41 1 − 10−5 4

1 2 1 − 3 − 10−3 1 3 − 14 − 10−4

3 1 3

− 14 − 10−4 1 4 1 − 5 − 10−5 1 5 − 16 − 10−6

. . .

. . .

. . .

4

5

− 41 1 − 10−5 4 − 51 1 − 10−6 5 − 61 1 − 10−7 6 . . .

1 5

− 16 − 10−6 1 6 1 − 7 − 10−7 1 7 − 18 − 10−8

. . .

··· ··· ··· ··· ··· ···

Clearly, each Ui exists and the sign of Ui is (−1)i+1 . The odd-indexed Ui are all positive and easily calculated: U1 = ln 2 = 1/(1 × 2) + 1/(3 × 4) + 1/(5 × 6) + · · · , U3 = 1 − ln 2 = 1/(2 × 3) + 1/(4 × 5) + 1/(6 × 7) + · · · . In general, * U2k−1 = (−1)k+1 ln 2 − 1 − =

∞  j=0

1 2

+ · · · + (−1)k /(k − 1)

1 , (k + 2 j)(k + 2 j + 1)

k ≥ 2.

Any U2k is negative and is given by ∞ 

U2k =

j=k+1

−

1 10−k . =− j 10 9

Note that on the one hand ∞ 

U2k = −

k=1

  ∞ 1 1  −k =− . 10 9 81 k=1

On the other hand, ∞ 

U2k−1 =

k=1

∞   k=1

1 1 + + ··· k(k + 1) (k + 2)(k + 3)

∞  ∞  ∞  1 1 ∞ 1 > > dt 4 4 j2 t2 n n=1 j=n

=

1 4

∞  n=1

1 = ∞. n

n=1



+

100

Convergence of lattice sums and Madelung’s constant

Since the sum of the positive terms diverges and the sum of the negative terms n Ui does not exist. Thus converges, it follows that limn→∞ i=1   ∞ ∞   diverges, ai j i=1

j=1

even though it satisfies Campbell’s conditions for convergence. Campbell goes on to claim that the analogous result holds for any dimension and that one could also prove convergence if one summed by expanding rectangles. Both these statements are unfounded. Remark 3: In light of the above, it appears that there is no simple proof in the literature of the convergence of any of the most elementary direct summation methods. That is why detailed proofs of Theorems 2.2 and 2.4 are given. Emersleben’s result (Theorem 2.3) indicates that a non-casual approach is justified. Remark 4: The proofs given of Theorems 2.2 and 2.4 are simple and intuitive, based as they are on the fact that the contribution of a basic unit cell to the sum always has the same sign as that of the nearest point in the cell to the origin. We have abstracted this property and have obtained quite general convergence results for multidimensional alternating series (see for instance Chapter 8). We will point out later that Theorems 2.2 and 2.4 also follow from the deeper considerations of the next section.

2.4 Integral transformations and analyticity Our purpose in this section is to establish a firm connection between the elementary direct summation methods discussed above and integral transformation methods, which are described by Glasser and Zucker in their survey article [20]. One major consequence of this connection is that we can give a definition of the Madelung constant, which has a firm mathematical foundation and is unique in a strong enough sense to indicate why diverse methods of performing the lattice sums lead to the same number. We begin with a general discussion of the analyticity of certain lattice sums in N -dimensional space. Of course N = 2 and 3 are the most interesting cases, but the general notation is just as convenient. For a complex number s, let Re s denote the real part of s and let   (−1)n N : n ∈ Z /{0} , A N (s) = ||n||2s where n = (n 1 , n 2 , . . . , n N ) ∈ Z N , (−1)n = (−1)n 1 +n 2 +···+n N , and ||n|| = (n 21 + n 22 + · · · + n 2N )1/2 . We also use the notations % & |n| = |n 1 |, |n 2 |, . . . , |n N | ∈ Z N

2.4 Integral transformations and analyticity

101

and, for m ∈ Z N , if n j ≥ m j for 1 ≤ j ≤ N .  If Re s > N /2 then a simple comparison test shows that n=0 1/||n||2s < ∞. So the elements of A N (s) are absolutely summable if Re s > N /2. Let    (−1)n n d N (2s) = : n ∈ Z /{0} . ||n||2s n≥m

Then d N (z) is a function of the complex variable z for Re z > N . In fact, it is a multidimensional zeta function, analytic on this domain. To see this define, for m ∈ Z N , with m > 0,    (−1)n n : n ∈ Z /{0} and |n| ≤ m . (2.4.1) d m (z) = ||n||z Then d m (z) is analytic for Re z > 0. For fixed δ > 0, if Re z > N + δ,   1 N N |d N (z) − d m (z)| ≤ : n ∈ Z /{l ∈ Z : |l| ≤ m} . ||n|| N +δ

(2.4.2)

The right-hand side of (2.4.2) can be made arbitrarily small by letting the minimal coefficient of m become large. Thus, on the region (Re z > N + δ), d N (z) is the uniform limit of analytic functions and is therefore analytic. Since δ > 0 is arbitrary we have established the following proposition. Proposition 2.1

The function d N (2s) is analytic in s for Re s > N /2.

Now comes the crucial step for the definition of the Madelung constant. The functions d N (2s) can be analytically continued to the region (Re s > 0). To accomplish this we follow the ideas of Glasser and Zucker [20] and introduce θ -functions and the Mellin transform. We need, in particular, θ4 (q) =

∞ 

2

(−1)n q n ,

0 ≤ q < 1.

n=−∞

Hence θ4 (e−t ) =

∞ 

(−1)n e−n t , 2

0 < t < ∞.

(2.4.3)

n=−∞

For a continuous function f (t) defined for 0 < t < ∞, bounded as t → 0, and decaying sufficiently fast as t → ∞, one can define a normalized Mellin transform Ms ( f ) for Re s > 0 by  ∞ Ms ( f ) =  −1 (s) f (t)t s−1 dt, 0

102

Convergence of lattice sums and Madelung’s constant

where  is the usual gamma function, given by  ∞ (s) = e−t t s−1 dt for Re s > 0. 0

 −1

(that is, 1/ ) are analytic functions on (Re s > 0). A useful Of course,  and property of the Mellin transform is that, for a > 0 and f such that its Mellin transform exists, Ms (τa f ) = Ms ( f )/a s , where τa f (t) = f (at) for all t > 0. In particular, Ms (e−at ) = 1/a s

for a > 0.

(2.4.4)

Consider now a truncation of the series for θ4 . For some positive integer m, let  n n2 N φm (q) = m n=−m (−1) q . If m ∈ N , say m = (m 1 , . . . , m N ), then let m = min{m 1 , . . . , m N }. We wish to approximate the N th power of θ4 with products of φm i . For 0 < t < ∞,1    N   N −t   N

−t   θ (e ) − |θ4 (e−t ) − φm i (e−t )| + φm i (e−t ) φ (e ) mi  4 ≤ i=1

i=1

−

N 

  N N   φm i (e−t ) = (bi + ai ) − ai ,

i=1

i=1

i=1

(2.4.5) where ai = φm i (e−t ) and bi = |θ4 (e−t ) − φm i (e−t )|. Note that 0 < ai , bi < 1 N N (bi + ai ) − i=1 ai , for i = 1, . . . , N , and the last expression in (2.4.5), i=1 represents the difference in volume between an N -box of side lengths bi + ai , i = N (bi + ai ) − 1, . . . , N and one of side lengths ai , i = 1, . . . , N . Clearly i=1 N N −1 . Now |θ (e−t ) − φ (e−t )| is the a < max{b : 1 ≤ i ≤ N }N 2 i i 4 m i=1 2 maximum bi and |θ4 (e−t ) − φm (e−t )| < 2e−m t . So (2.4.5) becomes   N   N −t  −t  N −m 2 t θ (e ) − φ (e ) . (2.4.6) mi  4  < N2 e i=1

N We also need the Mellin transform of i=1 φm i (e−t ) − 1, which is easily found using linearity and (2.4.4):     N  2 φm i (e−t ) − 1 = Ms (−1)n e−||n|| t Ms |n|≤m

i=1

=

 |n|≤m

(−1)n ||n||2s

= d (2s). m

(2.4.7)

1 The derivation following, up to the end of (2.4.6), contains minor mathematical errors and does not provide sharp estimates; subsequent appearances of N 2 N in Theorem 2.6 are also not sharp.

These issues are addressed in the Commentary at the end of the chapter: see (2.8.1).

2.4 Integral transformations and analyticity

103

The prime on the summation sign indicates that the n = 0 term is omitted. We are now ready for the main theorems of this section. Define F(s) = Ms [θ4N (e−t ) − 1] wherever it exists. Theorem 2.5 The Mellin transform F(s) of θ4N (e−t ) − 1 exists and is analytic for all s with Re s > 0. Furthermore F provides an analytic continuation of d N (2s) to the region (Re s > 0). Theorem 2.6 Re s > 0,

For any m ∈ Z N , m > 0, m = min{m i : 1 ≤ i ≤ N }, and |F(s) − d m (2s)| < N 2 N (Re s)/(m 2 Re s |(s)|).

Proof of Theorems 2.5 and 2.6 combined Re s > 0. Since

Let s be a complex number such that

0 ≤ 1 − θ4N (e−t ) ≤ N [1 − θ4 (e−t )] ≤ N e−t then



 θ N (e−t ) − 1 |t s−1 | dt ≤ N 4

∞ 0

Therefore, if Re s > 0 then  ∞ 0

(2.4.8)



∞

for all 0 < t < ∞,

e−t t Re s−1 dt = N (Re s).

0

N −t

θ4 (e ) − 1 t s−1 dt = F(s)

exists. Using (2.4.6) and (2.4.7), with m as in Theorem 2.6,   ∞ N   N −t  m −t  s−1  φm i (e ) |t | dt |(s)| |F(s) − d (2s)| ≤ θ4 (e ) − 0



i=1

∞

≤ N 2N

e

−m 2 t

t Re s−1 dt

0

= N 2 N (Re s)/m 2 Re s . Thus (2.4.8) holds. In turn, (2.4.8) implies that F(s) can be uniformly approximated by d m (2s) on any region of the form Rδ,M = {s : |s| < M and Re s > δ}. To see this, let K be an upper bound for the continuous function N 2 N (Re s)/|(s)| on the closure of Rδ,M . Then, for any  > 0 and any m such that m = min{m 1 , . . . , m N } > (K /)1/(2δ) , |F(s) − d m (2s)| < 

for all s ∈ Rδ,M .

Since  is arbitrary and d m is analytic, F is analytic on Rδ,M for any δ > 0 and 0 < M < ∞. Therefore F is analytic on (Re s > 0). Finally, it is now clear that F(s) agrees with d N (2s) if Re s > N /2. Thus F is an analytic continuation of d N .

104

Convergence of lattice sums and Madelung’s constant

In light of Theorem 2.5 we will drop the use of F and write d N (2s) = Ms [θ4N (e−t ) − 1]

for Re s > 0.

(2.4.9)

A rigorous mathematical definition can now be given for the Madelung constant. Definition 2.2 For a three-dimensional NaCl-type ionic crystal, the Madelung constant is the number d3 (1) = M1/2 [θ43 (e−t ) − 1]. Of course, this is the very number that has been approximated by many different methods over the years. We have just given a definition that avoids all the ambiguities of meaning that have existed. The uniqueness of analytic continuation explains the special significance of this particular sum of the elements of + * (−1)i+ j+k /(i 2 + j 2 + k 2 )1/2 : (i, j, k) ∈ Z3 /(0, 0, 0) . Formula (2.4.8) emphasizes the strong connection between integral transformation methods and direct summation methods. In fact it is worthwhile to formulate a corollary to Theorem 2.6, which gives explicit error bounds for a finite sum approximation to the Madelung constant. Corollary 2.1

Let m i > 0 for i = 1, 2, 3 and m = min{m 1 , m 2 , m 3 }. Then      12 (−1)i+ j+k d3 (1) − < .   2 2 2 1/2 (|i|,| j|,|k|) 0. In fact, expanding rectangles of any shape with sides parallel to the axes leads to d2 (2s). In Theorem 2.1 we showed that the method of expanding circles converged when s = 12 , but there is no reason to believe that d2 (1) is obtained unless one shows that the appropriate function is analytic. Using the notation of Section 2.2, let G(s) =

∞  (−1)n C2 (n)

ns

n=1

,

(2.5.1)

whenever the right-hand side converges. Then G(s) is the sum of the elements of As obtained by expanding circles. Theorem 2.7 The function G(s) exists and is analytic for Re s > 13 . Thus,  n 1/2 . G(s) = d2 (2s) if Re s > 13 ; in particular, d2 (1) = ∞ n=1 (−1) C 2 (n)/n Proof As in the proof of Theorem 2.1, let Bn = Bn = O(n 1/3− )

n

k=1 (−1)

kC

2 (k).

By (2.2.8),

for some  > 0.

(2.5.2)

Define G n (s) for all s with Re s > 0 by G n (s) =

n  (−1)k C2 (k)

ks

k=1

.

As in (2.2.9), 

Bn + Bk k −s − (k + 1)−s . s (n + 1) n

G n (s) =

(2.5.3)

k=1

By (2.5.2), if Re s ≥

1 3

then |Bn /(n + 1)s | → 0, uniformly in s. Note that

   |k −s − (k + 1)−s | = (−s)

k+1

k k+1



≤ |s|

  t −(s+1) dt 

t −(Re s+1) dt ≤ |s|k −(Re s+1) .

k

(2.5.4) So, for s ∈ R M = {z : Re z ≥ 13 , |z| ≤ M} with M a fixed positive number and 1 ≤ N ≤ N ,

106

Convergence of lattice sums and Madelung’s constant

   N  N −s

  −s  Bk k − (k + 1)  ≤ |Bk | |k −s − (k + 1)−s | 

k=N

k=N 

≤K

N 

k 1/3− |k −s − (k + 1)−s |

by (2.5.2)

k=N 

≤ KM

N 

k 1/3−−Re s−1

by (2.5.4)

k=N 1/3−−Re s

≤ KMN

≤ KMN − →0

as N , N  → ∞,

uniformly for s ∈ Rm . Thus the sequence of functions {G n (s)}∞ n=1 is uniformly Cauchy on R M and it converges uniformly to a limit function G(s). Furthermore, each G n (s) is analytic, so G(s) is analytic for s ∈ R M . Since M is arbitrary, G(s) exists and is analytic for all s with Re s > 13 . Remark 7: It is not known what the minimum non-negative β is, such that G(s) exists for all s with Re s > β (see Chapter 8). However, if we consider another method of summing the elements of As , we can get a very complete and illuminating analysis. This is the method of expanding diamonds. For each k = 1, 2, 3, . . . and complex s with Re s > 0, let δk (s) =

k 

(k − j)2 + j 2

−s

.

j=0

For each n = 1, 2, 3, . . ., let n (s) = 4

n  (−1)k δk (s) − 4X n (2s),

(2.5.5)

k=1

 where X n (s) = nk=1 (−1)k /k s . Note that n (s) counts the contributions within  k s the diamond |k| + | j| ≤ n. Now, limn→∞ X n (s) = ∞ k=1 (−1) /k = −α(s), and α is known to be analytic for Re s > 0. Therefore, in order to determine for which s the limit of n (s) exists and is analytic, it is sufficient to analyze ∞ k k=1 (−1) δk (s). We begin by establishing a number of facts about the sequence of δk . Proposition 2.2 √ √  1 k (a) We have limk→∞ δk ( 12 ) = 2 ln( 2+1). Thus ∞ k=1 (−1) δk ( 2 ) diverges. 1 (b) For real r ≥ 2 , δk−1 (r ) ≥ δk (r ), k = 2, 3, 4, . . .  1 k (c) The sum ∞ k=1 (−1) δk (s) exists and is analytic for Re s > 2 .

2.5 Back to two dimensions

107

Proof For (a), δk ( 12 )

k 

−1/2 (k − j)2 + j 2 = j=0

=

 k  1 j=0

k

1−

j k

2 +

 2 −1/2 j k



1

−1/2 (1 − t)2 + t 2 → dt 0 √ √ = 2 ln( 2 + 1).

as k → ∞

For the proofs of (b) and (c) it is very convenient to introduce the following function. For Re s ≥ 12 , let  π/4 s cos2s θ dθ − 1. V (s) = 2s(2 ) 0

V ( 12 )

= 0. If r ≥ Then V is continuous and   π/4 (2 cos2 θ )r dθ − 1 ≥ V (r ) = 2r 0

1 2

then

π/4 √

We proceed now to the proof of (b). Let r ≥ δk−1 (r ) − δk (r ) =

2 cos θ dθ − 1 = 0.

1 2

and k ≥ 2. Then

k ,  [(k − j)2 + ( j − 1)2 ]−r − [(k − j)2 + j 2 ]−r − k −2r j=1

=

k   j=1

≥

k   j=1



j j−1 j j−1

2r t dt − k −2r [(k − j)2 + t 2 ]r +1 2r t dt − k −2r [(k − t)2 + t 2 ]r +1

k

2r t dt − k −2r 2 + t 2 ]r +1 [(k − t) 0  1  2r u du −2r =k − 1 2 2 r +1 0 [(1 − u) + u ]   1/2   1  (v + 2 ) dv 1 − 1 v = u − = k −2r 2r 2 1 −1/2 2r +1 (v 2 + 4 )r +1    1/2 2r dv −1 = k −2r r +1 2 2 (v + 14 )r +1 0   π/4  −2r 2 r 2r (2 cos θ ) dθ − 1 (tan θ = 2v) =k =

(2.5.6)

0

0

= k −2r V (r ) ≥ 0.

108

Convergence of lattice sums and Madelung’s constant

That is, δk−1 (r ) ≥ δk (r ), for r ≥ 12 , k = 2, 3, 4, . . . To prove (c), let  > 0 and M < ∞ be arbitrary. Let + * R = z : Re z > 12 +  and |z| < M . For s ∈ R, let r = Re s. For k ≥ 2, we can write δk−1 (s) − δk (s) =

k−1 ,  [(k − j)2 + ( j − 1)2 ]−s − [(k − j)2 + j 2 ]−s j=1

+ (k − 1)−2s − 2k −2s k−1  j  2st dt = + (k − 1)−2s − 2k −2s . 2 2 s+1 j−1 [(k − j) + t ] j=1

Thus |δk−1 (s) − δk (s)| ≤

k−1  

j−1

j=1

≤ 2M

j

2|s|t dt + 3(k − 1)−2r [(k − j)2 + t 2 ]r +1

k−1  

t dt + 3(k − 1)−2r [(k − 1 − t)2 + t 2 ]r +1

j−1

j=1



j

k−1

t dt + 3(k − 1)−2r [(k − 1 − t)2 + t 2 ]r +1 0    1 u du −2r 2M = (k − 1) +3 2 2 r +1 0 [(1 − u) + u ]

= (k − 1)−2r (M/r )V (r ) + M/r + 3 ≤ (k − 1)−2r C, (2.5.7) = 2M

where C is the maximum of the continuous function (M/r )V (r ) + M/r + 3 for n 1 k=1 [δ2k−1 (s) − δ2k (s)] is an analytic function 2 +  ≤ r ≤ M. Now, for each n, of s for Re s > 12 , and    ∞ n 

k δ2k−1 (s) − δ2k (s) (−1) δk (s) = lim − n→∞

k=1

k=1

exists uniformly on R, by (2.5.7) and the Weierstrass M-test. Since  > 0 and M < ∞ are arbitrary, (c) has been established. We can now describe the behaviour of the diamond sums. Theorem 2.8 1, 2, . . ., let

For each complex number s with Re s > 0 and each n = n (s) =

n  l=1

 (−1)

l

 | j|+|k|=l

%

j +k 2

& 2 −s

 .

2.6 The hexagonal lattice

109

Then limn→∞ n (s) exists and is analytic for Re s > 12 . Although {n ( 12 )}∞ n=1 fails to converge,

lim n (r ) . (2.5.8) d2 (1) = lim r →1/2+ n→∞

Proof These claims all follow immediately from Proposition 2.2. Remark 8: Further analysis along the lines of Proposition 2.2 shows that although    n  % 2 & l 2 −1/2 j +k (−1) l=1

| j|+|k|=l

is divergent, it is Cesàro summable or Abel summable to d2 (1). The diamond sums provide a nice illustration of how a method of summing the elements of As can be analytic in s, for Re s large, but then with decreasing Re s this analyticity fails at a specific point. With the diamond sums it happens to be at 1 2 ; with expanding squares or rectangles it is at 0. It is not clear where it fails for expanding circles; it is at some point less than 13 . In three dimensions the method of expanding spheres fails at some point greater than 12 .

2.6 The hexagonal lattice As an illustration of what is obtained when one studies other crystal lattices in the above manner, we include a brief summary of results on the Madelung constant of a two-dimensional regular hexagonal lattice with ions of alternating unit charge. In order to obtain a tractable expression for the terms appearing in the lattice sum, choose a coordinate system with an angle of φ = π/3 between the positive axes. Then an arbitrary site in the lattice has coordinates (n, m), with n and m integers. A charge of +1, −1, or 0 is attached to that site in a regular fashion (see Fig. 2.2). By considering the two parallelograms indicated in Fig. 2.2, one can see that this charge may be expressed by q(n, m) =

+ 4* −sin nθ sin [(m −1)θ ]+sin [(m +1)θ ] sin [(n +1)θ ] , 3

θ=

2π . 3

The distance of the point (n, m) from the origin is given by  

(n, m) = (n + m/2)2 + 3(m/2) 1/2 . The set of numbers to be summed is then  2s * + Cs = q(n, m)/(n, m) : (n, m) ∈ Z2 /(0, 0)

for Re s > 0.

As before, the elements of Cs are absolutely summable for Re s > 1 and we require an analytic continuation of their sum to a region which includes s = 12 . Arguments like those used for the diamond sums will show that direct summation

110

Convergence of lattice sums and Madelung’s constant

Figure 2.2 The hexagonal lattice.

by expanding shells of hexagons will converge analytically for Re s > 12 and even have a limit as s approaches 12 from the right. However, for precise calculation purposes an analytic continuation via integral transform methods is far superior. Let    q(n, m) : (n, m) ∈ Z2 /(0, 0) , (2.6.1) H2 (2s) = |(n, m)|2s for Re s > 1. Then H2 is an analytic function of s and the series converges absolutely. Substituting the expression for q(n, m) and using elementary trigonometric identities yields √   cos[(m − n)θ ] 3   sin[(m − n)θ ] − , (2.6.2) H2 (2s) = 3 |(n, m)|2s |(n, m)|2s  indicates that the sum is over (n, m) ∈ Z2 /(0, 0). By symmetry conwhere siderations, the second term in the right-hand side of (2.6.2) is zero. Further manipulation of theta functions (using the modular equation of order 3) produces the rectangular sum   1 1   (−1)n+m  . (2.6.3) − H2 (2s) = (31−s − 1) 2 (n 2 + 3m 2 )s (n 2 + 3m 2 )s A theta function identity due to Cauchy (see Dickson [14]) and a Mellin transform yields H2 (2s) = 3(31−s − 1)ζ (s)L −3 (s),  −s where ζ (s) is the standard zeta function ( ∞ n=1 n ) and L −3 (s) = 1 − 2−s + 4−s − 5−s + 7−s − 8−s + · · · .

(2.6.4)

2.7 Concluding remarks

111

The formula (2.6.4) can also be deduced directly from (2.6.2) by using results in Section IV of Glasser and Zucker [20]. While the intermediate sums (2.6.3) and (2.6.4) are only analytic for Re s > 1, the standard continuation of the zeta function, (1 − 21−s )ζ (s) =

∞  (−1)n+1 n −s = α(s), n=1

gives H2 (2s) = 3(31−s − 1)(1 − 21−s )−1 α(s)L −3 (s).

(2.6.5)

The right-hand side of (2.6.5) is an analytic function of s for Re s > 0 and therefore (2.6.5) provides the required analytic continuation, which is necessary for the Madelung constant of this hexagonal crystal lattice: √ % & √ % & (2.6.6) H2 (1) = 3(1 − 3)(1 + 2)α 12 L −3 12 . This can be considered as a solution to this lattice sum problem, as both α( 12 ) and L −3 ( 12 ) can be rapidly calculated by known techniques. At s = 1, we have an exact result: π ln 3 (2.6.7) H2 (2) = − √ . 3

2.7 Concluding remarks We have investigated some fundamental properties of the multiply indexed series involved in the definition of the Madelung constant for an NaCl-type ionic crystal in two and three dimensions. We have provided elementary proofs that convergent series are obtained if the series is summed by letting the shape of a basic unit cell expand. The natural method of summing the effects of all ions within a fixed distance and letting the distance go to infinity leads to a convergent series in two dimensions but not in three dimensions. We have provided a unity to the concept of the Madelung constant by the use of the analytic continuation of a complex function. Thus, although the expression for the Madelung constant is conditionally convergent when summed by expanding squares (or cubes), other methods of summing will provide the same answer provided that they are ‘analytic’ in the correct sense. We have provided an analysis of the methods of expanding circles and expanding diamonds in two dimensions to illustrate this point. Perhaps the most important results are those in Section 2.4, relating the integral transformation methods and the direct summation methods. These integral transform methods are the most useful in practice as they lead to very rapidly convergent series.

112

Convergence of lattice sums and Madelung’s constant

In the course of these investigations we have encountered many curious facts, most of which are probably known to experts in the area. However, the formulas (2.6.6) and (2.6.7) seem to be unknown and to be of sufficient interest to have been included, at least as an illustration that the techniques of analytic continuation are applicable to other lattices.

2.8 Commentary: Improved error estimates Equation (2.4.5) and the subsequent proof of (2.4.6) contain errors resulting from the incorrect manipulation of inequalities. Here we provide a correct proof which simultaneously provides better error estimates for convergence. Without loss of generality, let m = m 1 ≤ m 2 ≤ · · · ≤ m N . We wish to N φm i (e−t )| for 0 < t < ∞. For notational convenience, bound |θ4N (e−t ) − i=1 N −t let θ := θ4 (e ) and let f i := φm i (e−t ). Observe first that 0 < θ, | f i | ≤ 1. Then observe, that, for |a|, |b| ≤ 1, |a n −bn | ≤ n|a−b| by factorization and the triangle inequality. By these observations and repeated use of the triangle inequality, we have   N θ − f 1 f 2 · · · f N      ≤ θ N − f 1N  + | f 1 |  f 1N −1 − f 2 f 3 · · · f N      ≤ N |θ − f 1 | +  f 1N −1 − f 2N −1  + | f 2 | f 2N −2 − f 3 f 4 · · · f N    ≤ N |θ − f 1 | + (N − 1)| f 1 − f 2 | +  f 2N −2 − f 3N −2    + | f 3 | f 3N −3 − f 4 f 5 · · · f N  .. . ≤ N |θ − f 1 | + (N − 1)| f 1 − f 2 | + (N − 2)| f 2 − f 3 | + · · · + | f N −1 − f N |. Now, as θ and each f i is an alternating series, it is easy to see that each absolute 2 value is less than 2e−m t . Summing the arithmetic progression above, we have established that N    2   N −t φm i (e−t ) < N (N + 1)e−m t . (2.8.1) θ4 (e ) − i=1

Equation (2.8.1) should be used in place of (2.4.6). In particular, in the statement (and proof) of Theorem 2.6, N 2 N should be replaced by N (N + 1), which is a tighter estimate for the error in the convergence.

2.9 Commentary: Restricted lattice sums The theta function method is easily adapted to yield possibly useful expressions for lattice sums subjected to various geometric restrictions. Here, this is illustrated

2.9 Commentary: Restricted lattice sums

113

by two examples of the form F(R) =



1 , Q(ρ)s

ρ

(2.9.1)

where ρ denotes a point of a crystal lattice in some dimension, lying in or on a sphere of radius R centred at one of the lattice points, which may or may not be included in the sum. The quantity Q represents a quadratic form, and s is usually a real parameter; they play an essential role in many areas of chemistry and physics [20] and in the mathematical study of the corresponding infinite sums (see the discussion earlier in this chapter). We finish with a third example, where the restriction is to a cone. Example 2.1 A restricted Lorenz–Hardy sum is given by 

F(R) =

(m,n)=(0,0)

(m 2

1 . + n 2 )s

(2.9.2)

For s = 12 this sum is required, for example, in studying the crystallization of a low-density electron gas on a metal surface [3] and for s = 0 (2.9.2) presents Gauss’ famous problem of counting lattice points enclosed in a circle [21] (in this case, centred at the lattice point at the origin). We proceed by means of a series of simple integral transformations. As usual, the Heaviside step function is denoted by θ . Thus, 

F(R) = =

1 (s)

(m,n)=(0,0)



∞ 0

dt t 1−s



θ (R 2 − m 2 − n 2 ) (m 2 + n 2 )s

θ (R 2 − m 2 − n 2 )e−t (m

2 +n 2 )

.

(2.9.3)

(m,n)=(0,0)

Throughout, it is assumed that s has values for which such operations are justified. Deficiencies later can usually be corrected by analytic continuation or judicious subtractions. Next, the step function is replaced by its representation as an inverse Laplace transform: 1 F(R) = 2πi(s)



c+i∞ c−i∞

dz R 2 z e z

 0

∞

dt t 1−s

 (m,n)=(0,0)

e−(z+t)(m

2 +n 2 )

. (2.9.4)

114

Convergence of lattice sums and Madelung’s constant

The double summation is easily carried out in terms of the Jacobi theta function [21], yielding   c+i∞ 1 dz R 2 z ∞ dt 2 e F(R) = [θ (0, q) − 1] 2πi(s) c−i∞ z t 1−s 3 0  ∞  c+i∞  R2 ∞ ∞ 4 dt dz uz   = du (−1)m q n(2m+1) , e (s) 0 t 1−s c−i∞ 2πi 0 n=1 m=0

(2.9.5) where q = exp[−(z + t)] and the theta function has been re-expressed by means of one of Jacobi’s famous q-identities [21],  (−1)m q n(2m+1) . (2.9.6) θ32 (0, q) = 1 + 4 m,n

The next step is to substitute y = z + t, which produces a shift in the positive constant c in the inverse Laplace transform. However, this can be undone by shifting the contour back as allowed by Cauchy’s theorem, since no singularities intervene. Therefore,  ∞   R2  dt −ut c+i∞ 4 du e dy euy (−1)m e−yn(2m+1) F(R) = s−1 (s) 0 t 0 c−i∞ m,n   2 ∞ ∞ R ∞ dt −ut 4  = (−1)m du e δ[u − n(2m + 1)], (2.9.7) 1−s (s) t 0 0 n=1 m=0

since the y-integral is a representation for the Dirac delta function. Performing the integrals now simply reverses the steps followed, and we have F(R) = 4

∞ 

(−1)

N (R,m) ∞  − n(2m + 1)] (−1)m 1 =4 , [n(2m + 1)]s (2m + 1)s ns

m θ [R

m=0

2

m=0

m=1

(2.9.8) where N (R, m) denotes the largest integer less than or equal to R 2 /(2m + 1). It should be noted that the m-sum is cut off at a finite upper limit < [R 2 ]. In the same way one finds that the restricted Madelung constant for the unit square lattice is M(R, s) =

 (m,n)=(0,0)

N (R,m) ∞  (−1)m+n (−1)m (−1)n = 4 . (2.9.9) s 2 2 s (2m + 1) ns (m + n ) m=0

n=1

We therefore have lim M(R, s) = M(∞, s) = −4(1 − 21−s )ζ (s)β(s).

R→∞

2.9 Commentary: Restricted lattice sums

115

Example 2.2 We now consider a restricted NaCl potential sum. By following the procedure above we find that S(R, s) = =

∞ 

(−1)l+m+n θ [R 2 − (l + 16 )2 − (m + 16 )2 − (n + 16 )2 ]

l,m,n=−∞  c+∞ 1

[(l + 16 )2 + (m + 16 )2 + (n + 16 )2 ]s  dz R 2 z ∞ dt e 2πi z t 1−s 0

(s) c−i∞  2 2 2 × (−1)l+m+n e−(z+t)[(l+1/6) +(m+1/6) +(n+1/6) ] . l,m,n

However, ∞ 

l −α(l+1/6)2

(−1) e

=q

1/24

l=−∞

∞ 

1/3 (−1) (2k + 1)q k

k(k+1)/2

k=0

where q = e−2α/3 . By noting that S(0, s) = 0 this can be manipulated into the form  ∞  c +i∞  R2 dt 1 du dy e−ut euy q 1/8 S(R, s) = (s) 0 t 1−s c −i∞ 0 ∞  × (−1)k (2k + 1)q k(k+1)/2 , k=0

e−2y/3 . Since

1 the y-integral yields δ[u − 13 k(k + 1) − 36 ], the remaining with q = integrations are straightforward and produce

S(R, s) = 3s

∞  (−1)k k=0

2k + 1 [k(k + 1) + 14 ]s

θ [3R 2 −

1 4

− k(k + 1)].

√ Since k√+ 12 + R 3 > 0 the argument of the Heaviside step function is positive if k < 3R − 12 , so we have the simple formula √

S(R, s) = 12s

3R−1/2  k=0

(−1)k . (2k + 1)2s−1

Now, for Re s > 12 , lim S(R, s) = 12s β(2s − 1)

R→∞

√ in agreement with [19]. However, S(R, 12 ) alternates between 2 3 and 0 as R increases, so that lim S(R, 12 )

R→∞

does not exist, whereas [19] lims→1/2+0 S(∞, s) =

√

3.

116

Convergence of lattice sums and Madelung’s constant

Example 2.3 We now consider a restricted conical sum. There is a recent literature on conical theta functions [18] of the form  2 2 q n +m , θ K (q) = (n,m)∈K

where K is a well-structured polyhedral pointed convex cone in the plane. The Mellin transform Ms [θ K (q)] provides another class of interesting ‘restricted’ lattice sums, of the form  1 . σ K (2s) = 2 (n + m 2 )s (n,m)∈K

2.10 Commentary: Other representations for the Madelung’s constant No closed form has ever been found for the Madelung constant. This is discussed at some length in [6], which gives a general discussion of the notion of a closed form. Many other representations have been proposed, as have various generalizations such as those in [11]. A particularly interesting study is due to Crandall [10]. Therein, one development starts with a lovely 1986 formula for θ43 (q) due to Andrews. This is θ43 (q) = 1 + 4

∞  (−1)n q n n=1

−2

1 + qn

∞  n=1,| j|0

(−1)n+m+ p − 6α 2 (s). (nm + mp + pn)s

(2.10.2)

% & Here α(s) = 1 − 21−s ζ (s) is again the alternating zeta (or Dirichlet eta) function. Crandall then proceeds to expand the triangular lattice sum in (2.10.2) in various ways. We comment in passing that there can be only 18 or 19 positive integers not of the form nm + mp + pn for n, m, p > 0 [5] and the existence of the nineteenth is ruled out by the Riemann hypothesis. The other eighteen are 1, 2, 4, 6, 10, 18, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462 (2.10.3) and the nineteenth if it existed would exceed 1011 . These are precisely the exceptional discriminants for which no indecomposable binary quadratic form exists [27]. The non-square elements of (2.10.3) are precisely twice the integers of type 2 listed in [4, p. 293] and 210 leads to Ramanujan’s most famous singular value, sent in his letter to Hardy and given in [4, (4.6.12)].

2.12 Richard Crandall and the Madelung constant for salt

117

We draw the reader’s attention also to the striking similarity in the structure of (2.10.2) and (6.3.1), which is a five-dimensional sum with a closed form. Andrews’ theta function representation of (2.10.1) also led Crandall to a remarkable integral:  1 2 π 1 + 3 r sin(2θ)−1 1  dθ dr.  M3 = − % &% 2 2 & π 0 0 1 + r sin(2θ)−1 1 + r cos θ 1 + r sin θ Using another expression for the Madelung constant in [10], Tyagi [28] obtains the fast converging series ( 18 )( 38 ) 1 1 log 2 4π − + √ + M3 (1) = − − √ 8 4π 3 2 2 π 3/2 2  (−1)m+n+ p (m 2 + n 2 + p 2 )−1/2 , −2

 m,n, p exp 8π m 2 + n 2 + p 2 − 1

(2.10.4)

where the constants before the sum alone give 10 digits of accuracy.

2.11 Commentary: Madelung sums, crystal symmetry, and Debye shielding In lattices and plasmas the presence of point-like electrostatic sources or charges can be accompanied by an exponential screening due to charges of opposite sign distributed in a cloud around them. This is called Debye screening or shielding and should be taken into account in physical applications of lattice sums in such situations. Two recent articles [24, 25] discuss the incorporation of Debye screening effects into the mathematical treatment of lattice sums for structures including cubic lattices (NaCl), face-centred cubic lattices, the body-centred cubic lattice (CsCl), and the diamond structure (ZnS). The Dirichlet series which lies behind much of this,  exp(2πiβr − αr 2 ) , (2.11.1) S(β, α) = 3 r ∈Z r2 is also discussed. From a more mathematical viewpoint, such screening corresponds to the Abel summation of a potentially divergent series [2, 9].

2.12 Commentary: Richard Crandall and the Madelung constant for salt A salt crystal is made up of alternating positive and negative electric charges based on a simple cubic lattice. The electrostatic energy of interaction of this array of charges was first calculated approximately by Madelung in 1918.

118

Convergence of lattice sums and Madelung’s constant

Essentially it requires the evaluation of the very slowly conditionally convergent triple sum α=



(−1)m+n+ p . (m 2 + n 2 + p 2 )1/2

(2.12.1)

This was the prototype for the evaluation of arrays of electric charges arranged on other crystal structures, all referred to as Madelung constants. However, any reference to the Madelung constant always implies the first example, α. No closed form for α has ever been found but Richard Crandall initiated novel attempts to provide better approximations to α with well-known constants of analysis, and these are reviewed here. Mathematician–physicist–inventor Richard Crandall It is with great sadness that the present authors announce the passing of their dear colleague Richard Crandall, who died on Thursday 20 December 2012, after a brief bout of acute leukaemia – the week before his 65th birthday on 29 December. Crandall had a long and colourful career. He was a physicist by training, studying with Richard Feynman as an undergraduate at the California Institute of Technology and receiving his Ph.D. in physics at the Massachusetts Institute of Technology under the tutelage of Victor Weisskopf, the Austrian–American physicist who discovered what is now known as the Lamb shift and who was one of the most influential post-war physicists. Richard often commented that he thought digitally in the fashion of an electrical engineer. Crandall was for many years at Reed College in Portland, Oregon, where he directed the Center for Advanced Computation. At the same time, he also worked for Next Computers (as Chief Scientist), and subsequently for Apple Computers (as Distinguished Scientist) where he was the head of Apple’s Advanced Computation Group. Crandall’s research spanned both the theoretical and practical realms: prime numbers, cryptography, data compression, signal processing, fractals, epidemiology, and, of considerable interest to the present authors, experimental mathematics. He held several patents. He produced many algorithms that are incorporated into Apple’s products including the iPod and the iPhone. The library of fast Fourier transforms that was produced by his Advanced Computation Group at Apple was described by a colleague of ours as ‘miraculously’ fast. He worked on image processing techniques for Pixar for 13 years, during the last two to remove artifacts that reportedly could only be seen on Steve Jobs’ personal projector, or to meet Jobs’ exacting personal requirements that raindrops should look like they did on celluloid (Richard’s tool was too natural for modern filmgoers). Indeed, Crandall was a close colleague of Steve Jobs for many years. Crandall was preparing to write a biography of Jobs, which sadly will not now be written.

2.12 Richard Crandall and the Madelung constant for salt

119

How we did not meet Unfortunately the author of this commentary section, John Zucker, never had the pleasure of meeting Richard Crandall in person. We established email connection over our joint interest in the Madelung constant α. It happened in this way. In 1987 Richard produced two papers on α both of which I was asked to referee. They were of course both very good and were accepted. However, in one paper I communicated (as a referee, anonymously) a proof of a conjecture he had made and, in the other, the result of a sum he had not known. From this he guessed very quickly who the anonymous reviewer was and we got into contact. Ever since α first appeared it has been subjected to much analysis, such as (a) finding ways to evaluate it rapidly or (b) to see whether it might be expressed in terms of other constants of analysis. It was clear from his paper with Delord [13] that the problem of evaluating α to many decimal places was easily solved by Richard, as he gave a value to some 50 decimal places. It is given here to 24 decimal places for future reference: α = −1.747564594633182190636212 . . .

(2.12.2)

Let me point out immediately that for all practical purposes, such as evaluating the lattice energy of salt, the first four figures would be enough to compare with any experimental value. But, just as one only requires π to a mere 35 digits to find the radius of the universe to the accuracy of the radius of a hydrogen atom, this does not stop mathematicians calculating it to 10 trillion digits. So, in the paper with Buhler [12], (b) was tackled. Richard and the Madelung constant Richard’s conception was to find an exact expression for α made up of a part which could be evaluated in terms of well-known constants of analysis plus an exponentially fast-converging residual sum. Two of these sums which arise here are   πr   1 (−1)m+n+ p πr  := S+ := , S+ m, n, p; t t r exp(πr /t) + 1   πr   1 (−1)m+n+ p πr  := S− := . S− m, n, p; t t r exp(πr /t) − 1   Here r := m 2 + n 2 + p 2 , and wherever appears it will denote summation over all indices from −∞ to ∞. Note that    πr   πr  2πr + S− = −2S− . (2.12.3) S+ t t t Richard found the following exact expression for α:  πr  π α = αC (t) = 4tC(t) − + 2S+ , 2t t

(2.12.4)

120

Convergence of lattice sums and Madelung’s constant

where (−1)m t  2 2 π m + 4t (n 2 + p 2 + q 2 ) (−1)m+n+ p 1  . = 2π m 2 + n 2 + p 2 + 4t 2 (q + 12 )2

C(t) : =

 Here  implies summation over all indices from −∞ to ∞, excluding the case when all the indices are simultaneously zero. The residual term 2S+ (πr /t) contributes very little to the value of α and the smaller t is, the less it contributes. So if C(t) could be evaluated it would provide an approximation to α. Now C(t) involves finding a four-dimensional sum. Lattice sums are often evaluated using properties of theta functions and because there are many more theta function identities available in four than in three dimensions, they are often more easily evaluated than three-dimensional sums. This indeed proved to be the case and the following evaluations (which are analytic continuations) were provided in [12]:  √     1 2 2 − 2 1 1 3 log 2 1 1 C 4 = , C 2 =√ , β 8 , C(1) = − + 8 4π π 2 where β(x) is the central beta function defined by β( p) := B( p, p) = So, for t = 14 , αC ( 14 ) = C( 14 ) − 2π + 2S+3

  1 4

 2 ( p) . (2 p)

 √   2 2 − 2 1 β 8 − 2π + 2S+3 14 , = π (2.12.5)

and we have

 √ 2 2 − 2 1 β 8 − 2π = −1.747523 . . . , π with the residual series 2S+ (4πr ). Comparing this with (2.12.2) gives agreement to 4 decimal places, good enough for any practical application. A full account of Richard’s struggle with α is given in his masterly presentation in [10]. Tyagi’s work This probably inspired another investigator, Sandeep Tyagi, to a further attack on α. In this work he naturally came into contact with Richard, who communicated his work to the author. It encouraged me to find a value for C(t) for t < 14 , namely  √ √ √   3 2 − 6 3 + 6 6 1 1 √ β 8 . = C 24 2π

2.12 Richard Crandall and the Madelung constant for salt

121

So, using this in (2.12.4), one has  √ √ √ √ 3 2 − 6 3 + 6 6 1 6 β 8 − = −1.7475621 . . . , 2π π √ with residual 2S+ ( 24πr ). This gives agreement to 5 decimal places with (2.12.2). However, this is trivial compared with what Sandeep had achieved in [28]. Sandeep found another exact expression for α, similar in structure to (2.12.4) but different. It was  πr  π − S− , (2.12.6) αT (t) = T (t) − 6t t where T (t) :=

2t  (−1)m+n+ p . π m 2 + n 2 + p 2 + 4t 2 q 2

Sandeep was able to evaluate T (t) for some values of t; thus   T

1 2

1 log 2 , =− − 2 π

  T

1 4

1 1 log 2 +√ . =− − 4 2π 2

Thus, for t = 1/4, αT ( 14 ) = − 14 −

2π log 2 1 +√ − − 2S− (4πr ). 2π 3 2

(2.12.7)

Very shrewdly, he then took the average of (2.12.5) and (2.12.7) to obtain  √   1 log 2 1 4π 2 2−2 1 αC T = − + √ − − + β − 2S− (8πr ). (2.12.8) 8 2 2 3 4π 2π 8 It is seen that the residual sum is even smaller. (This result might have been found directly from a relation discovered later between C(t) and T (t). This is 4tC(t) + T (t) = 2T (t/2),

hence

2T ( 18 ) = T ( 14 ) + C( 14 ),

which gives (2.12.8) immediately.) When we evaluate the constants in (2.12.8) we obtain −1.7475645947, in which the error from α is just 1 in the tenth decimal place. A remarkable agreement! Further refinement As a kind of tribute to Richard, I suggested to Sandeep that we might try to improve on this. One way of doing this would be to find C(1/8), but we were not able to do it. However, we could look at evaluating certain terms of 2S− (8πr ).

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Convergence of lattice sums and Madelung’s constant

Thus Sandeep considered finding this for m = n = 0, n = p = 0, and p = m = 0. We then get three contributions that are all the same and finish with 2S− (1, 0, 0; 8πr ) = −12

∞  p=1

(−1) p . p(e8π p − 1)

(2.12.9)

Now we use the following result in [29]: ∞  p=1

1 4 πc (−1) p − log  , =− 2π pc 12 12 kk p(e − 1)

(2.12.10)

where k and k  are the values of the modulus and complementary modulus of the complete elliptic integrals K and K  of the first kind, which are found when K  /K = c. Here c = 4, and (2.12.9) can be evaluated to give √ 9 45 log( 2−1)−6 log(21/4 +1)− log 2. (2.12.11) 2 8 When this is added to the constants of (2.12.8) we obtain −1.747564594633175, which agrees with the α value given by (2.12.2) to 1 in the 14th decimal place. Thus encouraged, the next set in 2S− (8πr ) was evaluated. There are three terms of the form, (m = 0, n = p), (n = 0, m = p) and ( p = 0, m = n), leading to 2S− (1, 0, 0; 8πr ) = 4π +

∞ √ 1 24  √ . 2S− (1, 1, 0; 8 2πr ) = − √ 8 2π p + 1) 2 p=1 p(e

(2.12.12)

To evaluate this we require a result in [29]: ∞ 

1

p=1

p(e2π pc + 1)

=−

2K 3 kk  πc 1 − log , 12 6 π3

(2.12.13)

√ with c = 4 2 in (2.12.12). The evaluation of (2.12.12) is somewhat involved but the result is  √ √ 4(1 − b)2 2b(1 + b2 ) 2S− (1, 1, 0; 8 2πr ) = 8π + 2 2 log (1 + b)4   23/4 (1 + b)2 β 18 √ , (2.12.14) + 6 2 log 64π √ where b := (2 2−2)1/4 . Adding this contribution to (2.12.8) and (2.12.11) gives −1.7475645946331821917 and the disagreement with α occurs only in the 18th decimal place. Finally let us add the contribution of 2S− (8πr ) when m = n = p. In this case, ∞ √ (−1) p 16  √ . 2S− (1, 1, 1; 8 3πr ) = − √ 3 p=1 p(e8 3π p − 1)

(2.12.15)

References

123

√ and here c = 4 3; 2S− (1, 1, 1) may be evaluated to give √ 16 (1 − d)4 4√  3 log , (2.12.16) 2S− (1, 1, 1; 8 3πr ) = π+ 3 9 32(1 + d)2 2d(1 + d 2 ) √ √ 1 where d := (( 3 + 1)/ 8) 2 . When this is added to the constants of (2.12.8), (2.12.11), and (2.12.14) we obtain −1.74756459463318219063622 where the agreement with α is now up to 22 decimal places. It is clear how further terms may be evaluated. Conclusion The Madelung constant was only a tiny part of Richard Crandall’s many-faceted interests. Far more important was his contribution to computational number theory amongst many other accomplishments. The brute force approach used here of evaluating terms of the residue sums to add more constants of analysis to the value of α would, I am sure, not have won the approval of Richard. He would have preferred a more subtle approach on the lines of Sandeep’s derivation of (2.12.8). Still, I think he would have been amused by the results given here.

References [1] J. S. Blakemore. Solid State Physics. Saunders, Philadelphia, 1969. [2] D. Borwein and J. M. Borwein. Some exponential and trigonometric lattice sums. J. Math. Anal. Appl., 188:209–5218, 1994. [3] D. Borwein, J. M. Borwein, R. Shail, and I. J. Zucker. Energy of static electron lattices. J. Phys. A: Math. Gen., 21:1519–1531, 1988. [4] J. M. Borwein and P. B. Borwein. Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity. Wiley, New York, 1987. [5] J. M. Borwein and K.-K. S. Choi. On the representations of x y + yz + zx. Exp. Math., 9:153–158, 2000. [6] J. M. Borwein and R. E. Crandall. Closed forms: what they are and why we care. Not. Amer. Math. Soc., 60(1):60–65, 2013. [7] E. L. Burrows and S. F. A. Kettle. Madelung constants and other lattice sums. J. Chem. Ed., 52:58–59, 1975. [8] E. S. Campbell. Existence of a well defined specific energy for an ionic crystal – justification of Ewald’s formulae and of their use to deduce equations for multipole lattices. J. Phys. Chem. Solids, 24:197, 1963. [9] A. N. Chaba and R. K. Pathria. Evaluation of a class of lattice sums in arbitrary dimensions. J. Math. Phys., 16:1457–1460, 1975. [10] R. E. Crandall. New representations for the Madelung constant. Exp. Math., 8:367– 379, 1999. [11] R. E. Crandall. The Poisson equation and ‘natural’ Madelung constants. 2012. Preprint.

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[12] R. E. Crandall and J. P. Buhler. Elementary function expansions for Madelung constants. J. Math. Phys., 20:5497–5510, 1987. [13] R. E. Crandall and J. F. Delord. The potential within a crystal lattice. J. Math. Phys., 20:2279–2292, 1987. [14] L. E. Dickson. An Introduction to the Theory of Numbers. Dover, New York, 1957. [15] O. Emersleben. Math. Nachr., 4:468–480, 1951. [16] H. Evjen. On the stability of certain heteropolar crystals. Phys. Rev., 39:675–687, 1932. [17] P. Ewald. Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys., 64:253–287, 1921. [18] A. Folsom, W. Kohnen, and S. Robins. Conic theta functions and their relations to theta functions. 2011. Preprint. [19] P. J. Forrester and M. L. Glasser. Some new lattice sums including an exact result for the electrostatic potential within the NaCl lattice. J. Phys. A, 15:911–914, 1982. [20] M. L. Glasser and I. J. Zucker. Lattice sums. In Theoretical Chemistry, Advances and Perspectives (H. Eyring and D. Henderson, eds.), vol. 5, pp. 67–139, 1980. [21] R. K. Guy. Gauss’ lattice point problem. In Unsolved Problems in Number Theory, 2nd edition. Springer-Verlag, New York, 1994. [22] G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers, 4th edition. Clarendon, Oxford, 1960. [23] K. Højendahl. K. Dan. Vidensk. Selsk. Mat. Fys. Medd., 16:135, 1938. [24] S. Kanemitsu, Y. Tanigawa, K. Tsukada, and M. Yoshimoto. Crystal symmetry viewed as zeta symmetry. Chapter 9 in Zeta Functions, Topology and Quantum Physics. Springer, 2005. [25] S. Kanemitsu and H. Tsukada. The Legacy of Alladi Ramakrishnan in the Mathematical Sciences: Crystal Symmetry Viewed as Zeta Symmetry, II. Springer, 2010. [26] C. Kittel. Introduction to Solid State Physics. Wiley, New York, 1953. [27] M. Peters. The Diophantine equation x y + yz + x z = n and indecomposable binary quadratic forms. Exp. Math., 13:273–274, 2004. [28] S. Tyagi. New series representation for the Madelung constant. Progr. Theoret. Phys., 114:517–521, 2005. [29] I. J. Zucker. Some infinite series of exponential and hyperbolic functions. SIAM J. Math. Anal., 15:406–413, 1984.

3 Angular lattice sums

We discuss here the topic of angular lattice sums, i.e., those which depend on the angle or angles between the vector linking the origin to lattice points and the coordinate axes. This topic is an old one, dating back to an 1892 paper by Lord Rayleigh [26], but is curiously disconnected from the main thread of investigations into lattice sums, as surveyed by Glasser and Zucker [9]. We begin with a brief account of the history of the sums and go on to give an account of some of their more recently discovered properties. We use the latter topic to discuss how properties and formulae for lattice sums may be discovered with the aid of modern symbolic algebra packages such as Mathematica or Maple. Chief among the properties of the angular lattice sums in two dimensions that we describe is their relationship to the Riemann zeta function; selected sums obey the celebrated Riemann hypothesis.

3.1 Optical properties of coloured glass and lattice sums The technology of colouring glass by adding to the melt appropriate metals is an old one, dating back to the time of the ancient Greeks and Romans and predating the modern field known as plasmonics. Michael Faraday proposed a model of atoms as being like tiny metal particles and so opened up the question as to what could be inferred about the properties of atoms from the interaction of light with various solids. This question was addressed by various investigators, adopting the approach of supposing that light interacted with the atoms to give them a dipole moment due to charge separation and that the optical response of the solid could then be deduced by adding up the periodically arranged response charges. The most important of the resulting expressions are the Clausius–Mossotti formulae [4, 20], the Lorentz–Lorenz law [13, 14], and the Maxwell–Garnett equation [7]. An informative account of the history of this approach may be found

126

Angular lattice sums

in the review article by R. Landauer [12]. Rayleigh commented in his paper that the treatments of Lorenz and Lorentz, as well as those of previous authors, take into account only dipole–dipole interactions between particles. His purpose was to show that this is an approximation, in that for concentrated systems of particles accuracy is only achieved when higher-order effects such as those of induced quadrupoles and octupoles are taken into account. In order to do this, Rayleigh considered both electrostatic problems in two dimensions, involving rectangular and square lattices of cylinders, and also a simple cubic lattice of spheres. The two-dimensional problem is to find a potential function V (x, y) which satisfies Laplace’s equation inside and outside each cylinder and which obeys continuity of the function and its scaled normal derivative at cylinder boundaries:   ∂ Vint (x, y)  ∂ Vext (x, y)  σ = Vint (x, y)|∂C = Vext (x, y)|∂C ,   , ∂n ∂n ∂C ∂C (3.1.1) where ∂/∂n denotes the normal derivative on the boundary of a typical cylinder and σ denotes a relative transport coefficient such as electrical conductivity or permittivity (its value inside each cylinder being divided by its value between cylinders). We write the potential expansion around the central cylinder in the form of a multipole expansion in polar coordinates (r, θ ): Vext (r, θ ) =

∞  

An r n +

n=1

Bn  cos nθ. rn

(3.1.2)

The choice of a cosine dependence is appropriate if there is an applied external field along the O x axis and the sum runs only over odd values of n. The coefficients An (and corresponding internal multipole coefficients Cn ) can be related to the coefficients Bn using the boundary coefficents (3.1.1). If E 0 denotes the magnitude of the external applied field then the coefficients Bn are obtained by solving a linear equation called the Rayleigh identity. This identity states that the part of the expansion (3.1.2) which does not diverge at the origin, i.e., the part involving the coefficients An , is due to sources on all other cylinders in the array and the sources at infinity. It may be written as 



∞

B2n−1  (2n + 2m − 3)! S2n+2m−2 B2m−1 = E 0 δn,1 , + (2m − 2)! a 4n−2 m=1 (3.1.3) where a denotes the cylinder radius and the lattice sum of order 2n is (2n − 1)!

1+σ 1−σ

S2n =

 R p =(0,0)

cos 2nθ p . R 2n p

(3.1.4)

3.1 Optical properties of coloured glass and lattice sums

127

In this lattice sum, p runs over all the points in the lattice apart from the origin, the polar coordinates of the pth point being R p and θ p . Given that the lattice sums can be evaluated, the system of equations (3.1.3) permits the evaluation of the multipole coefficients B2n−1 . The effective transport coefficient for the array of cylinders is then given by 2π B1 , (3.1.5) σe f f = 1 − El where El denotes the local field in the vicinity of the central cylinder. One subtle question concerning the Rayleigh multipole method is that of the evaluation of the dipole sum S2 . This sum for a square lattice takes the form S2 =



p12 − p22

( p1 , p2 )=(0,0)

( p12 + p22 )2

,

(3.1.6)

and thus is conditionally convergent. Rayleigh dealt with this problem by summing over a region much more elongated along the x axis than the y axis. Perrins et al. [23] showed that the shape-dependent part of the summand in (3.1.7) below could be absorbed into the expression relating El to E 0 . Another approach is to introduce a charge-neutralising background distribution, as followed in Chapter 8 and in Poulton et al. [24]. Rayleigh’s approach to the evaluation of the other lattice sums is quite sophisticated and is of sufficient interest to reproduce an outline of it here. He commences with the sums for the square lattice and notes that they are zero except if the order is even. The sum of order 2n can then be written for a square lattice of unit period:  1 . (3.1.7) S2n =  (m + im)2n  (m,m )=(0,0)

He then notes that sin(ξ − imπ ) is zero when ξ = m  π + imπ , m  being positive, negative, or zero. It then follows that, for a suitable constant A,      ∞  ξ ξ ξ 1− 1− . sin(ξ − imπ ) = A 1 − imπ imπ + m  π imπ − m  π  m =1

(3.1.8) Putting ξ = 0, we find that A = − sin(imπ ). Rayleigh expands the left-hand side of (3.1.8), divides by A, and takes the logarithm to obtain   ξ log [cos ξ − cot(imπ ) sin ξ ] = log 1 − imπ     ∞   ξ ξ + log 1 − . + log 1 − imπ + m  π imπ − m  π  m =1

(3.1.9)

128

Angular lattice sums

He then combines (3.1.9) and its equivalent with m replaced by −m to obtain

  sin2 ξ ξ2 log 1 − = log 1 − (imπ )2 sin2 imπ     ∞   ξ2 ξ2 log 1 − + log 1 − . + (imπ + m  π )2 (imπ − m  π )2  m =1

(3.1.10) This identity can be expanded in a power series in ξ on both sides and the coefficients equated to give expressions for the sums of various orders. The result for S2 is ∞  1 π2 − 2π 2 S2 = = π, (3.1.11) 3 (sinh mπ )2 m=1

where the hyperbolic function sum is obtainable from tables or given by a symbolic algebra package. Rayleigh’s result for S4 is  ∞   π4 1 2 4 + S4 = 2π . (3.1.12) + 45 sinh4 π m 3 sinh2 π m m=1 The hyperbolic function series is rapidly convergent: five terms are sufficient to give the result 3.1512120021538976, accurate to 2 parts in 1014 (the exact value is ( 14 )8 /(960π 2 ); see Section 4.8). Given the lattice sums, closed forms may be derived for the effective conductivity or permittivity of a square lattice of circular cylinders of radius a. The higher-order Rayleigh formula for the square array of circular cylinders takes into account the multipole coefficients B1 , B3 , B5 , and B7 , and with T = (1 − σ )/(1 + σ ) is −1  0.305827 f 4 T 0.013362 f 8 − 2 f, (3.1.13) σeff = 1 − T + f − 2 T T − 1.402958 f 8 where f = πa 2 , for a unit lattice spacing, is the area fraction of cylinders in the array. The numerical values in (3.1.13) take into account the values of the lattice sums S2 , S4 , S8 , S12 . Their incorporation into the formula for σe f f results in an expression which takes into account better the multipole interactions between cylinders in densely packed composites [23]. Rather than continuing with Rayleigh’s argument, we will now follow a more modern development due to Mityushev [18] and Mityushev and Adler [19].

3.2 Lattice sums and elliptic functions Mityushev [18] and Mityushev and Adler [19] consider a general two-dimensional lattice of cylinders, the lattice being generated by fundamental translations in the

3.2 Lattice sums and elliptic functions

129

complex plane ω1 = α > 0 and ω where Im ω = α −1 , so that the parallelogram cell thus defined has unit area. These authors denote the lattice points m 1 α + m 2 ω by a linear ordered set e j , with j running from zero to infinity and e0 = 0. With this notation, the lattice sums of the previous section become  e−2n (3.2.1) S2n = j , j

where the primed sum runs over the whole lattice apart from e0 . Let h = exp(−π/α 2 ). Then Mityushev and Adler [19] give the following expressions for low-order lattice sums:   ∞  π 2 1  mh 2m −8 , (3.2.2) S2 = α 3 1 − h 2m m=1   ∞  m 3 h 2m 1  π 4 1 + 16 S4 = , (3.2.3) 3 α 15 1 − h 2m m=1

and 1  π 6 S6 = 15 α



∞

 m 5 h 2m 2 − 16 63 1 − h 2m

 .

(3.2.4)

m=1

In addition, they give the following recursion formula, which, commencing with S4 and S6 , gives all higher-order sums: S2k =

k−2  3 (2m − 1)(2k − 2m − 1)S2m S2(k−m) . (3.2.5) (2k + 1)(2k − 1)(k − 3) m=2

Applying the recursion formula (3.2.5) yields lattice sums above S4 and S6 as combinations of these two: 3 2 5 1 S , S10 = S4 S6 , S12 = (18S43 + 25S62 ), 7 4 11 143 3S4 (33S43 + 100S62 ) 5S6 (783S43 + 275S62 ) , S18 = , = 2431 46189 3S 2 (2178S43 + 12125S62 ) , ... S20 = 4 508079

S8 = S16

(3.2.6)

The results (3.2.6) apply to any two-dimensional lattice and give the sums as functions of the quotient 1/α of the height and the horizontal side length α of the parallelogram unit cell. They simplify in the case of a square lattice, for which S6 = S10 = S14 = · · · = 0, so that (3.2.6) becomes 3 2 18 3 99 4 S , S , S , S12 = S16 = 7 4 143 4 2431 4 6534 4 620730 6 S , S , = S24 = ... 508079 4 151915621 4

S8 = S20

(3.2.7)

130

Angular lattice sums

This list, combined with the value quoted above for S4 , gives highly accurate values for the square lattice sums up to S24 : (l, Sl ) = ((4, 3.1512120021538976), (8, 4.2557730353651895), (12, 3.9388490128279705), (16, 4.0156950330250245), (20, 3.9960967531762903), (24, 4.000976805303839)).

(3.2.8)

Rayleigh [26] was able to obtain expressions equivalent to those for S4 and S8 and also highly accurate values for them. These lattice sums arise in the Taylor series for the Weierstrass ρ(z), ζ (z), and σ (z) functions [29]. We will now discuss these series and the relationship between the function log σ (z) and Green’s functions. The Taylor series for log σ (z) is log σ (z) = log z −

∞  S2n 2n z . 2n

(3.2.9)

n=2

This function has a logarithmic singularity at z = 0, and also at all the lattice points e j , since      z z z2 1− exp σ (z) = z + 2 , (3.2.10) ej ej 2e j j where the prime over the product symbol denotes the omission of (0, 0). The Weierstrass invariants for the general lattice are g2 = 60S4 , g3 = 140S6 . The function σ (z) is related to the Jacobi theta function θ1 (z) by   S2 2 θ1 (z/α) (3.2.11) exp z . σ (z) = α  θ1 (0) 2 We show the behaviour of the real and imaginary parts of this function in Fig. 3.1. The real part of log σ (z) is a periodic function with a logarithmic singularity at the origin. The imaginary part of log σ (z) has a branch-cut singularity along the negative real axis in the plots generated in Mathematica and also Maple, according to their conventions used. The imaginary part of log σ (z) has a linear gradient along the sides of the unit cell parallel to the y axis, the values there related by Im{log[σ (0.5 + i y)]} − Im{log[σ (−0.5 + i y)]} = π(y − sign(y)).

(3.2.12)

The other two Weierstrass functions are given by ∞

1  S2n z 2n−1 ζ (z) = − z

(3.2.13)

n=2

and ∞

ρ(z) =

 1 + (2n − 1)S2n z 2n−2 . 2 z n=2

(3.2.14)

3.3 A phase-modulated lattice sum Re log σ

131 Im log σ

0

3 2 1 0 –1 –2 –3

–1 –2 –3 –4 –5 –0.4

–0.4 –0.2 y

–0.4

–0.4 –0.2

–0.2 0

0 0.2

0.2 0.4

y

x

–0.2 0

0 0.2

0.2 0.4

0.4

x

0.4

Figure 3.1 Plots of the real (left) and imaginary (right) parts of the Weierstrass function log σ (z), as z ranges over the unit cell of the square lattice.

These Taylor series are derived in an obvious way by differentiating the corresponding series (3.2.9) for log σ (z).

3.3 A phase-modulated lattice sum We next consider a phase-modulated lattice sum, following a paper by Glasser [8]. This sum and aspects of its treatment will be of importance in much of the rest of this chapter. It consists of a sum over the reciprocal lattice corresponding to a rectangular lattice in two dimensions with periods a and b along the x and y axes, respectively. The reciprocal lattice is then also rectangular, and consists of the vectors u v  u, v = 0, ±1, ±2, . . . (3.3.1) k = 2π , , a b Then the phase-modulated sum is =

 eik·S k=0

ks

.

(3.3.2)

Here S is an arbitrary spatial vector lying in the direct lattice; k·S is a dot product. This lattice sum may also be regarded as a Green’s function, where the operator is the Laplacian raised to the complex power s/2 (using the ideas of fractional calculus or distribution theory):    eik·S = i s (ab) δ(S − R p ) − 1 , (3.3.3) s/2  = i s k=0

Rp

132

Angular lattice sums

where R p = ( p1 a, p2 b) denotes a general lattice vector in the direct lattice. Here the Poisson summation formula has been used in going from the first form of the right-hand side in (3.3.3) to the second. Note that the integral of the righthand side over any unit cell in the direct lattice is zero. In this sense, the Green’s function  is source-neutral, as discussed by Poulton et al. [24]. Glasser [8] starts with the expression =

 a s 2π

 (u,v)=(0,0)

e2πi(ux+vy) [u 2 + (a/b)2 v 2 ]s/2

(3.3.4)

and takes its Mellin transform, to give 

∞

(s) = (a/2π ) dt t s/2−1 [T (t) − 1], 0  T (t) = exp{−t[u 2 + (a/b)2 v 2 ] + 2πi(ux + vy)}, s

(3.3.5)

u,v

where the sum in T extends over all values of u, v. The next step is to apply the Poisson summation formula, in this case equivalent to the Jacobi theta function transformation, 

exp(−tu 2 + 2πiux) = (π/t)1/2

u



exp[−(π 2 /t)(u + x)2 ].

(3.3.6)

u

This gives, after separating out the term v = 0 from the double sum, T (t) − 1 =



e−tu

2 +2πiux

u=0

+ (π/t)1/2



e−(π

2 /t)(u+x)2

e−a

2 v 2 t/b2 +2πivy

.

u;v=0

The double sum here is over all integers u, but over v excluding v = 0. Consequently, ⎧  a s ⎨ e2πiux = 2π ⎩ us u=0

⎫  ⎬ π 1/2  2πivy ∞ 2 2 2 2 + e dt t (s−3)/2 e−π (u−x)/t e−a v t/b . (3.3.7) ⎭ (s) 0 u;v=0

We next use Hobson’s integral, giving the integral in (3.3.7) as a Macdonald function, in our case of complex order and real argument: 

∞ 0

dt t s−1 e− pt e−q/t = 2(q/ p)s/2 K s [2( pq)s/2 ],

(3.3.8)

3.3 A phase-modulated lattice sum

133

assuming p > 0, q > 0, and Re s > 0. This gives the expression for  in its final general form: =

   a s  e2πiux 2π 1/2  π b|u + x| (s−1)/2 + 2π us (s) |v| u=0 u;v=0    2πa 2πivy . (3.3.9) |v||u + x| e ×K (s−1)/2 b

The Macdonald functions in (3.3.9) converge exponentially to zero as soon as the magnitudes of the integers v and u become sufficiently large that the argument exceeds by a factor of around 2 the magnitude of the order, |s − 12 |/2. This makes this form of expansion suitable for evaluations anywhere in the complex plane of s. The expression (3.3.9) simplifies when s is an even integer, as the Macdonald √ functions of half-integer order reduce to the factor exp(−z)/ z multiplied by a polynomial. The most important case is, of course, that for s = 2, when  can be regarded as a periodic Green’s function for the Laplace equation. For this case,  K 1/2 (z) =

π 2z

1/2

e−z ,

so that (3.3.9) becomes ⎡ ⎤  a 2  e2πiux  1 π b ⎣ = e−2π(a/b)|v||u+x|+2πivy ⎦ . + 2π a |v| u2 u=0

(3.3.10)

(3.3.11)

u;v=0

We take the term u = 0 out of the double sum and then combine positive and negative values of u and v in the same sums. We introduce the quantities q = e−π(a/b) ,

α = y + i(a/b)x,

(3.3.12)

and obtain =

∞ ∞ a 2  cos 2π ux ab  1 2πiαv (e + + c.c) 4π v 2π 2 u2

+

u=1 ∞  ∞ 

ab 4π

u=1 v=1

v=1

q 2uv v

(e2πiαv + e−2πiαv + c.c.),

(3.3.13)

where c.c. denotes the complex conjugate. The v sums can be carried out simply, since ∞  xv = − log(1 − x), v v=1

(3.3.14)

134

Angular lattice sums

so that =

∞ a 2  cos 2π ux ab |1 − e2πiα | − 2 2 2π 2π u

−

u=1 ∞ 

ab 2π

log |(1 − q 2u e2πiα )(1 − q 2u e−2πiα )|.

(3.3.15)

u=1

The first sum is a well-known Fourier series [25], and so (3.3.15) becomes =

a2 2 ab (x − x + 16 ) − log(2e−πax/b | sin π α|) 2 2π  ∞ ∞    ab   2u 4u  − 1 − 2q cos 2π α + q  . log    2π u=1

(3.3.16)

u=1

The infinite product can be written in terms of the Jacobi theta function θ1 [29]: ∞  

 1 − 2q 2u cos 2π α + q 4u = 21/3 cosec z q −1/6

u=1

θ1 (z, q) . [θ1 (0, q)]1/3

(3.3.17)

This leads to the final result

   θ1 (π α, q)  a 2 2 ab ab , (3.3.18) x − log 2 − log   2 6π 2π [θ1 (0, q)]1/3   n−1/2 q (n+1/2)2 e(2n+1)i z . Here we have corrected where θ1 (z, q) = ∞ n=−∞ (−1) a typographical error in the coefficient of the second term on the right-hand side in [8]. The function (x, y) is shown in Fig. 3.2. It resembles that for the real part of log σ (z) shown in Fig. 3.1, except in so far as (x, y) is source-neutral whereas log σ (z) is not. =

3.4 Double sums involving Bessel functions We now consider a class of double sums involving Bessel functions, which in some cases have exact polynomial representations. This class of sums was investigated [22] for a square lattice of period d: 1  Jl (K h ξ ) i4mψh e . (3.4.1) Sl,4m,n (ξ ) = n d K hn h=0

Here, l, m = 0, 1, 2, . . . and n = 2, 3, . . . are integers, ξ = d(x, y) with 0 ≤ x, y ≤ 1 and Kh = 2π(h 1 , h 2 )/d is a general reciprocal lattice vector, which has the polar representation Kh = (K h , ψh ). If ξ is set equal to zero and n = 4m, these sums reduce to scaled versions of the sums considered by Rayleigh for the square lattice. We will next exhibit their connection with Glasser’s function .

3.4 Double sums involving Bessel functions

135

Φ

0.5 0.4 0.3 0.2 0.1 0 –0.1 –0.2 –0.4 –0.2

–0.4 –0.2

y

0

0 0.2

0.2

x

0.4

0.4

Figure 3.2 Plot of the Green’s function (x, y) for the Laplace equation (s = 2) as (x, y) ranges over the unit cell of the square lattice.

To do this, we note that =

 e2πi(ux+vy)/d ks

k=0

,

(3.4.2)

so that, if we replace k by K h (cos ψh , sin ψh ) and expand the exponential term using the generating equation for the Bessel functions J p , we find =

∞  

ip

K h =0 p=−∞

J p (ξ K h )ei p(θ−ψh ) , K hs

(3.4.3)

where (x, y) = ξ(cos θ, sin θ ). For the square lattice, the sums over K h in (3.4.3) are zero unless p is a multiple of 4, when they are real, so that for s = n we have =d

n

∞ 

S4 p,4 p,n (ξ )e4i pθ .

(3.4.4)

p=−∞

Also, the sums S4 p,4 p,n (ξ ) are even under the replacement of p by − p, so that   ∞   = d S(0, 0, n) + 2 S4 p,4 p,n (ξ ) cos(4 pθ ) . n

(3.4.5)

p=1

Thus, the Bessel function sums (3.4.1) over the reciprocal lattice (at least if s is an integer not smaller than 2) can be seen as the coefficients in the angular Fourier series in the direct lattice of the Green’s function .

136

Angular lattice sums 3.4.1 Angular-independent sums

We will now indicate the derivation of explicit polynomial solutions for the subset of Sl,4m,n (ξ ) for which they exist. We start with the case m = 0, for which the sums which have polynomial expressions have l + n even. (For l + n odd, the expression for Sl,0,n (ξ ) consists of a term in ξ n−2 and a non-terminating hypergeometric function [22].) The derivation relies on the construction of raising and lowering operators for the lattice sums. The raising operators rely on the integrals [28]  a Jl+1 (ba) x l+1 Jl (bx) d x = a l+1 (3.4.6) b 0 and



a

x −l+1 Jl (bx) d x =

0

J−l+1 (ba) bl−2 − a −l+1 . l−1 b 2 (l)

From these we find a result which increases both l and n:  ξ ηl+1 Sl,0,n (η) dη = dξ l+1 Sl+1,0,n+1 (ξ ),

(3.4.7)

(3.4.8)

0

while for n − l > 0 we obtain a result which lowers l and raises n:  ξ 1 d −l+2  η−l+1 Sl,0,n (η) dη = l−1 − dξ −l+1 Sl−1,0,n+1 (ξ ). 2 (l) (K h d)n−l+2 0 K h =0

(3.4.9) The series occurring on the right-hand side of equation (3.4.8) is of course familiar by now: s  s   1 4 L . (3.4.10) = ζ −4 (K h d)s (2π )s 2 2 K h =0

The two lowering operators follow from the following derivative relations for Bessel functions [28]:   1 l+1 x Jl+1 (bx) = x l+1 Jl (bx), (3.4.11) b and



1 −l+1 x Jl−1 (bx) b



= −x −l+1 Jl (bx),

(3.4.12)

where the prime denotes differentiation with respect to x. Hence, the operation which lowers both l and n is   (3.4.13) Sl−1,0,n−1 (ξ ) = dξ −l ξ l Sl,0,n (ξ ) , and that which lowers n and increases l is

  Sl+1,0,n−1 (ξ ) = −dξ l ξ −l Sl,0,n (ξ ) .

(3.4.14)

3.4 Double sums involving Bessel functions

137

The final recurrence relation of use is that derived from the recurrence relation for Bessel functions: 2l Jl (z) = Jl−1 (z) + Jl+1 (z), z which gives



2d Sl+2,0,n (ξ ) = (l + 1) ξ

(3.4.15)

 Sl+1,0,n+1 (ξ ) − Sl,0,n (ξ ).

(3.4.16)

To start the use of the recurrence relations, we use the results (3.3.18) and (3.4.5), which enable us to evaluate the lowest-order sum, S(0, 0, 2). Comparing these two, we find that    θ1 (π α, q)  1 ξ2 1   log 2 − log   , (3.4.17) S0,0,2 (ξ ) = 2 − 6π 2π 4d [θ1 (0, q)]1/3  a.i.p. where the notation ‘a.i.p.’ means that we are to take the part of the expression that is independent of the angle θ . Note that log |θ1 (π α, q)|a.i.p. = log

ξ + log 2π θ1 (0, q), 2d

leading to 1 ξ S0,0,2 (ξ ) = − log + ω0 + 2π 2d



ξ 2d

(3.4.18)

2 ,

(3.4.19)

where ω0 = −

1 [3 log π + 4 log 2 + 2 log θ1 (0, e−π )]  −0.318895593319287. 6π (3.4.20)

The (ξ/2d)2 term in (3.4.19) is equivalent to the ‘jellium’ used by [11] and others, and fulfils the function of removing the net charge from the Green’s function. Substituting (3.4.19) into (3.4.5) (with n = 2) and using the relation (3.4.29), we obtain another expression for  [24], free of the shape-dependent sum S2 which has caused difficulties in previous formulations. Commencing with the result (3.4.19), we can use the raising and lowering operators from (3.4.8), (3.4.9), (3.4.13), and (3.4.14) to determine explicit expressions for any sum Sl,0,n (ξ ) with l + n even. Examples of the results of this process are        ξ ξ ξ 1 ξ 3 1 1 log + ω0 + + S1,0,3 = − , (3.4.21) 2π 2d 2d 4π 2d 2 2d S2,0,4 (ξ ) = −

1 4π



ξ 2d

2 log

   2   1 ξ ξ 3 1 ξ 4 + ω0 + + , 2d 2 8π 2d 6 2d (3.4.22)

138 and 1 S0,0,4 (ξ ) = 2π

Angular lattice sums 

ξ 2d

2

  2    λ ξ ξ 1 1 ξ 4 + log − ω0 + − , 2d 2π 2d 4 2d 24π 2 (3.4.23)

where in the last of these we have used ζ (2) = π 2 /6 and L −4 (2) = λ  0.915965594177219, λ being the Catalan constant. The last two results that we give are for m = 0:   1 ξ 2 1 − (3.4.24) S2,0,2 (ξ ) = 4π 2 2d and 1 S3,0,3 (ξ ) = 8π



ξ 2d



1 − 6



ξ 2d

3 .

(3.4.25)

3.4.2 Angular-dependent sums For m = 0, the recurrence relations (3.4.8), (3.4.13), (3.4.14), and (3.4.16) remain unaltered. We have to reconsider the relation (3.4.9), which becomes  ξ eimψh d −l+2  η−l+1 Sl,m,n (η) dη = l−1 − ξ −l+1 d Sl−1,m,n+1 (ξ ), 2 (l) (K h d)n−l+2 0 K h =0

(3.4.26) provided that n − l ≥ 0. The double sum in (3.4.26) was shown in Movchan et al. [21] to take the following value for the square lattice:     e4imψh e4imψh 1 1 2π (4m) σs δs,2 , = = − (K h d)s (2π )s (2π )s 4m (h 21 + h 22 )s/2 K =0 (h ,h )=(0,0) h

1

2

(3.4.27) where the Kronecker delta contribution in the last term arises since the following limit has been taken:  eimψh e2πiR·(h 1 ,h 2 )/d σs(m) = lim . (3.4.28) R→0 (h 21 + h 22 )s (h ,h )=(0,0) 1

2

This exceptional contribution is related to the sum S2 discussed in the first section of this chapter. Thus, the required recurrence relation for angular sums is    ξ d −l+2 2π (4m) −l+1 σn−l+2 − δl,n η Sl,4m,n (η) dη = l−1 4m 2 (l)(2π )n−l+2 0 − ξ −l+1 dSl−1,4m,n+1 (ξ ),

(3.4.29)

for n − l ≥ 0. It will be observed then that all recurrence relations preserve the angular order 4m of the sums.

3.4 Double sums involving Bessel functions

139

The simplest form for an angular sum occurs when l = 4m and n = 2, and is (4m)

S4m,4m,2

σ = 4m 16π m

 4m ξ . d

(3.4.30)

(4m) Here the σ4m are just the S4m discussed in the first two sections of this chapter. Another useful starting point for the application of the recurrence formulae is

S2,4m,2 (ξ ) =

    2m−1 m  (2m + 1)s (−2m + 1)s (4m) ξ 2s+2 1 ξ 2 σ2s+2 − . 2π s!(s + 2)! d 8 d s=0

(3.4.31) Here, (x)n = (x + n)/ (x) denotes the Pochhammer symbol. For example, if m = 1 then we have     1  (4) π  ξ 2 4 (4) ξ 4 σ2 − S2,4,2 (ξ ) = − σ4 , (3.4.32) π 2 2d π 2d 2 (4) (4) (4) where σ2  4.0784511611614. Let us define σ2 = σ2 − π/2. Then, using (3.4.32) as our starting point, the recurrence relations give the following explicit forms for m = 4 sums:    3   ξ ξ 12 1 2 2 (4) ξ 5 (4) (4) − σ2 + σ4 , (3.4.33) σ S1,4,3 = 2d π 2d π 2d (2π )2 2  3  5 ξ ξ 1 2 1 S3,4,3 = − σ4(4) , (3.4.34) σ2(4) 3π 2d π 2d  2  4   ξ ξ 1 2 1 2 1 (4) ξ 6 (4) (4) σ S2,4,4 = − + , (3.4.35) σ σ 2d 3π 2 2d 2π 4 2d 8π 2 2  2   ξ 1 2 22 6 (4) ξ 4 (4) (4) σ S0,4,2 = − + , (3.4.36) σ σ π 2 2d π 4 2d 4π 2 2  2  4   ξ ξ 1 1 2 1 2 2 (4) ξ 6 (4) (4) (4) σ S0,4,4 = σ − + − , σ σ 2d 2π 2 2d 3π 4 2d 16π 4 4 4π 2 2 (3.4.37)    3  5 ξ ξ ξ 1 2 1 1 2 (4) (4) (4) − S1,4,5 = σ4 + σ2 σ2 4 2 2d 2d 6π 2d 16π 8π  7 1 (4) ξ σ − . (3.4.38) 6π 4 2d By the time n has reached 5, the sums (3.4.1) over the reciprocal lattice are easily evaluated by direct summation, enabling the easy verification of formulae of the type just presented.

140

Angular lattice sums

3.5 Distributive lattice sums We next consider lattice sums which are connected with the theory of distributions or generalized functions, following the discussion in the paper by McPhedran et al. [17]. Such distributive lattice sums are associated with sets of local sources at each lattice point, whose amplitudes are linked by a phase factor which varies in plane-wave fashion from one lattice point to the next. This phase factor arises very often in solid state physics, where it is called the Bloch factor. The lattice sums that we discuss are associated with solutions of the Helmholtz equation with wavenumber k, and we denote the Bloch vector which determines the phase factor by k0 . If R p denotes the pth lattice vector, in polar form (R p , φ p ) in two dimensions, the basic lattice sums are given by  (1) Hl (k R p )eilφ p eik0 ·R p . (3.5.1) Sl (k, k0 ) = p=0 (1)

Here the term Hl (kr )eilφ is a solution of the Helmholtz equation which behaves like an outgoing cylindrical wave for large r , and whose source is located at r = 0. (1) The Hankel function of the first kind Hl (kr ) is the sum of a Bessel function Jl (1) and a Neumann function Yl , so that Hl (kr ) = Jl (kr ) + iYl (kr ), and in terms of the last functions we write Sl = SlJ + i SlY . We first evaluate the lattice sums SlJ (k, k0 ). The Poisson summation formula in two dimensions is  2   2π e−is·R p = δ(s − Kh ), (3.5.2) d p h

where we are dealing with a square lattice of period d and the reciprocal lattice vectors Kh have Cartesian components (2π h 1 /d, 2π h 2 /d), h 1 and h 2 being integers. We put s = k − k0 , so that 1+



e

−i(k−k0 )·R p

 =

p=0

2π d

2 

δ(k − k0 − Kh ).

(3.5.3)

h

With θ = arg k, we have from the Bessel function expansion e

−ik·R p

=

∞ 

(−i)l Jl (k R p )eil(φ p −θ) ,

(3.5.4)

l=−∞

which combined with (3.5.3) gives 1+

∞  l=−∞

l −ilθ

(−i) e

 p=0

Jl (k R p )e

ilφ p ik0 ·R p

e

 =

2π d

2 

δ(k − k0 − Kh ).

h

(3.5.5)

3.5 Distributive lattice sums

141

We define phased reciprocal lattice vectors Qh by Qh = k0 + Kh and put Q h = |k0 +Kh |, θh = arg Qh . The right-hand side of (3.5.5) can then be rewritten to give  2  ∞   δ(k − Q h )δ(θ − θh ) 2π . (−i)l e−ilθ Jl (k R p )eilφ p eik0 ·R p = 1+ d k p=0

l=−∞

h

(3.5.6) Multiplying equation (3.5.6) by exp(inθ ) and integrating over θ from 0 to 2π yields 2πi l  SlJ (k, k0 ) = −δl,0 + δ(k − Q h )eilθh . (3.5.7) kd 2 h

SlJ

The are thus distributive in nature, their distributive nature being concentrated around the lines Q h = k in reciprocal space. These are called light lines in the literature on photonic band gap materials and correspond to the dispersion relation of plane waves. We next consider the lattice sums SlY . We obtain an expression for these by comparing two different expressions for a quasi-periodic Green’s function satisfying  δ(r − R p )eik0 ·R p . (3.5.8) (∇ 2 + k 2 )G(r) = −2π d 2 p

These are a spatial form, G(r) =

iπ  (1) H (k|r − R p )|eik0 ·R p , 2 p 0

(3.5.9)

and a spectral form, G(r) = −

2π  eiQh ·r . d2 Q 2h − k 2 h

(3.5.10)

We expand the Hankel function terms in (3.5.8) using Graf’s addition theorem [1], after separating out the term for p = 0. We expand the exponential term in the spectral form using (3.5.4). The two terms are compared by integrating over the angular direction of the arbitrary vector r from 0 to 2π , to give (1

Sl Jl (kr ) = −H0 (kr )δl,0 − or (1

Sl Jl (kr ) = −H0 (kr )δl,0 −

4i l+1  Jl (Q h r )eilθh d2 Q 2h − k 2 h

(3.5.11)

  1 4i l+1  Jl (Q h r )eilθh 1 . + 2Q h Qh + k Qh − k d2 h (3.5.12)

142

Angular lattice sums

The representation (3.5.12) has one sum which is non-singular on the right-hand side but a second sum which has singularities at values h such that Q h = k. These singularities are treated using the Plemelj formula, which expresses them as a combination of a delta function and a Cauchy principal value:   1 1 = iπ δ(x) + P . (3.5.13) x − i0 x We then obtain Sl Jl (kr ) = −J0 (kr )δl,0 − iY0 (kr )δl,0 −

  1 4i l+1  Jl (Q h r )eilθh 1 . ) + P + iπ δ(k − Q h 2Q h Qh + k Qh − k d2 h

(3.5.14) Comparing equations (3.5.14) and (3.5.6), we find SlY Jl (kr )

  1 1 4i l+1  Jl (Q h r )eilθh +P . = −Y0 (kr )δl,0 − 2 2Q h Qh + k Qh − k d h

(3.5.15) This equation shows that the SlY are singular at light lines, with first-order poles there which are to be treated numerically using the Cauchy principal value. The representation (3.5.15) is of formal use in deriving properties of the lattice sums SlY . An alternative form may be derived using the free parameter r and integrating over it to improve the convergence of the series for large values of Q h [3]. The result after m integrations is

  m 1  (m − n)! 2 m−2n+2 Y Sl Jl+m (kr ) = − Ym (kr ) + δl,0 π (n − 1)! kr − 4i l

 h

n=1

k Qh

m

Jl+m (Q h r )eilθh . Q 2h − k 2

(3.5.16)

3.6 Application of the basic distributive lattice sum The expression for the distributive lattice sums (3.5.7) may be used to generate expressions for double sums related by the Poisson summation formula. We can multiply this expression by any real or complex function of the wavenumber k:  p=0

f (k)Jl (k R p )eilφ p eik0 ·R p = − f (k)δl,0 +

2πi l  f (Q h ) δ(k − Q h )eilθh . Qh d2 h

(3.6.1)

3.6 Application of the basic distributive lattice sum

143

We then eliminate the delta functions on the right-hand side of (3.6.1) by integration over k:    ∞ f (k)Jl (k R p )dk eilφ p eik0 ·R p p=0

0



∞

=− 0

 f (k) dk δl,0 +

2πi l  f (Q h ) ilθh e . Qh d2

(3.6.2)

h

To apply the general formula (3.6.2), we need f (k) to be such that the two integrals over k are evaluable in closed form, either as ordinary integrals or in the sense of generalized functions. Three other applications of the formula are given in [17] but we will focus here on only one, which involves complex powers of k. We rely on results obtained using the methods of the theory of distributions from Chapter 7 of the book by D. S. Jones [10] and, in particular,  ∞  ∞ ( 12 β)!2β+2 π |x|β e−iα·x dx = 2π r β+1 J0 (αr ) dr = , (3.6.3) (− 12 β − 1)!α β+2 −∞ 0 valid when β = 2m and β = −2 − 2m (m = 0, 1, 2, . . .). Hence, if s = 2m − 1 or −1 − 2m,    ∞  12 + 12 s 2s  r s J0 (αr ) dr =  . (3.6.4) 0  12 − 12 s α 1+s By differentiation under the integral sign in (3.6.4) and the use of Bessel function recurrence relations, we obtain Weber’s integral,    ∞  12 + 12 l + 12 s 2s  r s Jl (αr ) dr =  , (3.6.5) 0  12 + 12 l − 12 s α 1+s provided that 1 + l ± s is not a negative even integer. If we put f (s) = k 2s−1 in (3.6.2) and use Weber’s integral, the result is straightforward for l = 0: ⎛ ⎞ l ilθ0  eik0 ·R p eilφ p  eilθh e 2πi ⎠. = 2 (1 − s + 12 l) ⎝ 2−2s + 22s−1 (s + 12 l) R 2s d k0 Q 2−2s p h p=0

h=0

(3.6.6) The result is not so straightforward for l = 0 because of the term  2s ∞  ∞ k k 2s−1 dk = , 2s 0 0

(3.6.7)

which is undefined at either the lower or the upper limit. This difficulty is dealt with in the original article [17] by the introduction of raising and lowering operators, partial derivative combinations with respect to the components of the vector

144

Angular lattice sums

k0 . When these operate, they either reduce or increase the integer l by one, while decreasing 2s to 2s −1. Through the use of these operators, it can be demonstrated that the nugatory term (3.6.7) is to be replaced by zero. Using this, we obtain ⎛ ⎞  eik0 ·R p  1 2π 1 ⎠. 22s−1 (s) = 2 (1 − s) ⎝ 2−2s + (3.6.8) R 2s d k0 Q 2−2s p h p=0

h=0

One valuable application of the equations (3.6.6) and (3.6.8) is to deduce functional equations for lattice sums. If we let the length of the vector k0 tend to zero, then the sums over the pairs (R p , φ p ) for the square array may be identified with those over the pairs (Q h , θh ) (we use here the fact that the square lattice is identical, apart from a rescaling, to its reciprocal lattice). When |k0 | → 0, for the square lattice the angular lattice sums in (3.6.6) tend to zero unless l is divisible by 4. If we define the following dimensionless sum, (2l + s)  e4ilφ p (2l + s) = π 2l+s (R p /d)2s π 2l+s



( p1 + i p2 )4l , 2 + p 2 )s+2l ( p 1 2 p=0 ( p1 , p2 )=(0,0) (3.6.9) we find that it satisfies for all integers l the functional equation G 4l (s) =

G 4l (s) = G 4l (1 − s).

(3.6.10)

For a fixed value of the integer l, G 4l (s) is an analytic function of the complex variable s, and the functional equation shows that G 4l (s) is real on the critical line Re s = 12 . This fact will be important in the discussion of the location of its zeros which follows. We also note the correspondence between the G 4n (s) and the sums S4n mentioned above in connection with the work of Lord Rayleigh: S4n = G 4l (2n).

3.7 Cardinal points of angular lattice sums We continue the discussion of angular lattice sums using elements of the article [15]. There, the following notation is introduced for a dimensionless form of the lattice sums occurring in equation (3.6.6): m (k0 ) = d 2s σ2s

 exp(imφ p ) exp(ik0 · R p ) R 2s p

(3.7.1)

p=0

and μl2s (k0 ) =

 exp(ilθh ) exp(ilθ0 ) l = + ρ2s (k0 ). 2s (d Q h ) (k0 d)2s

d Qh

(3.7.2)

3.7 Cardinal points of angular lattice sums

145

l (k ) is continuous as k → 0, while μl (k ) is certainly well defined as Here ρ2s 0 0 2s 0 k0 → 0 if Re s < 0. These sums are connected by (3.6.8): l (k0 ) = π 2s−1 i l σ2s

( 12 l + 1 − s) ( 12 l + s)

μl2−2s (k0 ).

(3.7.3)

We now replace k0 by the dimensionless quantity κ = (k0 d)/(2π ) and investigate the limit for l = 0 as s → 1 by putting s = 1 + 12 δ, where δ → 0: π 1+δ (− 12 δ)

σ20 (k0 ) = lim

δ→0

(1 + 12 δ)

0 (κ, θ0 )]. [eδ ln κ + ρ−δ

(3.7.4)

0 (k ) which may be derived by expanding the factor We use an expression for ρ2s 0 2s 1/Q h in a Taylor series for small κ:

ρ2−2s (κ, θ0 ) =

(0) σ2−2s

+

 ∞   (l + 1 − s) 2 l!(1 − s)

l=1

(0)

κ 2l σ2l+2−2s

∞ ∞  cos 4nθ0  (4n + q + 1 − s)(q + 1 − s) (4n) σ2q+4n+2−2s κ 2q+4n . +2 q!(q + 4n)! [(1 − s)]2 n=1

q=0

(3.7.5) Here we have introduced the notation (m) m σ2s = lim σ2s (k0 ),

(3.7.6)

k0 →0

and we recall that for the square lattice m must be divisible by 4 for a non-zero (4n) value to occur, while σ4n = S4n as defined in Section 3.2. We need to keep in (3.7.4) only terms which remain non-zero as δ → 0. The prefactor is of the form 2π/δ, and so we need to keep in the single sum over l in (3.7.5) only the term for l = 1. In the double sum, we need to retain only q = 0. Hence

∞ 2  −2π cos 4nθ δ 0 (0) (0) S4n κ 4n . 1 + δ ln κ + σ−δ + κ 2 σ2−δ − δ σ20 (κ, θ0 ) = lim δ→0 δ 4 4n n=1

(3.7.7) (0) Since δ2s has a simple pole at s = 1, we have (0)

σ2−δ 

−2π +D δ

(3.7.8)

& δ% C + ln 2π , 2

(3.7.9)

and (0)

σ−δ = −1 +

146

Angular lattice sums

giving σ20 (κ, θ0 ) = 2π ln κ − π(C + ln 2π ) + π 2 κ 2 +

∞ π  cos 4nθ0 S4n κ 4n . (3.7.10) 2 n n=1

In these expressions,

( 14 )4  0.783188785414, C = ln 8π 2   π (3.7.11) D = π 2γ + ln − C  2.5849817596. 2 In this section and in Chapter 4 we will be interested in the evaluation of the lattice sums σ20 (κ, θ0 ) at points in the Brillouin zone (the central unit cell in the reciprocal lattice), where (κx , κ y ) = (κ cos θ0 , κ sin θ0 ) takes rational values, of the form (m x /n, m y /n). The particular question we will consider now is the location of the special or cardinal points where the σ20 (κ, θ0 ) take simple closed forms consisting of a single term similar to that encountered in the Lorenz–Hardy result:

(0)

σ2s = 4ζ (s)L −4 (s).

(3.7.12)

Using the formula (3.6.7), the closed forms for the σ20 (κ, θ0 ) at the cardinal points also give simple expressions for the displaced lattice sums  1 . (3.7.13) Sm x ,m y ;n (2s) = 2 [(h 1 + m x /n) + (h 2 + m y /n)2 ]s h 1 ,h 2

We can locate the cardinal points by exploiting their analytic form, which must (0) consist of a product of σ2s with an algebraic prefactor [15]. It is easy to derive 0 (κ , κ ) following the derivation used in Macdonald function series for the σ2s x y 0 (κ , κ ) over Section 3.3 [15]. These can be used to construct contour plots of σ2s x y the Brillouin zone. We note that if s is chosen to correspond to a zero of either 0 (κ , κ ) must have a zero at every cardinal point. This ζ (s) or L −4 (s) then σ2s x y procedure is used in Fig. 3.3, where s is chosen to coincide with the first zero of L −4 (s) (at s  12 + 6.020948905i). The cardinal points evident in Fig. 3.3 were isolated by methods other than numerical strategies apart from the innermost points, which occur at (κx , κ y ) = 1 3 3 1 , ± 10 ) and (κx , κ y ) = (± 10 , ± 10 ). The forms of the lattice sums at the (± 10 cardinal points of the square lattice are: S0,1;2 (2s) = 22s+1 (1 − 2−s )L 1 (s)L −4 (s), 0 σ0,1;2 (2s) = 23−2s (1 − 2s−1 )L 1 (s)L −4 (s),

(3.7.14)

S1,1;2 (2s) = 2s+2 (1 − 2−s )L 1 (s)L −4 (s), 0 σ1,1;2 (2s) = 23−s (1 − 2s−1 )L 1 (s)L −4 (s),

(3.7.15)

S1,1;4 (2s) = 4 (2 − 1)L 1 (s)L −4 (s), s

s

0 σ1,1;4 (2s) = −22−3s (2s − 2)L 1 (s)L −4 (s).

(3.7.16)

3.8 Zeros of angular lattice sums

147

0.6

0.5

0.4

0.3

0.2

0.1 0.1

0.2

0.3

0.4

0.5

0.6

0 (κ , κ ) for s equal to the first zero of L (s), as Figure 3.3 Plot of the lattice sum σ2s x y −4 (κx , κ y ) ranges over the first quadrant of the Brillouin zone of the square lattice. The dots

represent cardinal points.

These expressions may be derived easily from the basic form (3.7.12). The cardinal point related to n = 5 was reported by Zucker [30], the lattice sum being evaluated by him using theta function techniques: S1,2;5 (2s) = (5s − 1)L 1 (s)L −4 (s),

0 σ1,2;5 (2s) = (−1 + 51−s )L 1 (s)L −4 (s). (3.7.17) Note the pattern in these sums: the prefactor in S is non-zero at s = 1, while the first-order zero of the prefactor in σ 0 cancels the first-order pole of ζ (s) there. For n = 10 we have

S1,3;10 (2s) = (5s−1 − 1)(2s−1 − 1)L 1 (s)L −4 (s), 0 σ1,3;10 (2s) = (1 − 21−s )(1 − 51−s )L 1 (s)L −4 (s).

(3.7.18)

Note that, for s → 1, both lattice sums tend to zero since the prefactor has a 1 3 , ± 10 ) and (κx , κ y ) = double zero there. In fact, the points (κx , κ y ) = (± 10 3 1 0 (± 10 , ± 10 ) all lie on the contour σ2 (κx , κ y ) = 0. The data of Fig. 3.3 pertains to the region where κ > 0.1. The function 0 (κ , κ ) = 0 is highly oscillatory near κ = 0, being dominated by the first σ2s x y term κ 2s−2 . However, this first term has modulus 1/κ for Re s = 12 , and so there are no cardinal points in the region where κ is non-zero and small.

3.8 Zeros of angular lattice sums The celebrated Riemann hypothesis affirms that all the non-trivial zeros of ζ (s) lie on the critical line Re s = 1/2. It has received extensive numerical confirmation

148

Angular lattice sums

but has yet to be proved analytically. The generalized Riemann hypothesis affirms that, for every Dirichlet L-function associated with a Dirichlet character, once again the non-trivial zeros lie on the critical line. Thus, assuming the generalized (0) Riemann hypothesis to hold, the basic sum σ2s will have all its non-trivial zeros on Re s = 12 , as will the sums σ20 (κx , κ y ) at all cardinal points. It is an open question whether comparable statements can be made for other points in the Brillouin zone or for other sums. However, some very significant results have been obtained, which should motivate further investigations. The results pertain to the following sums, which already have been introduced in (3.6.9): C(1, 4m; s) =

  cos 4mθ p , p 1 2 p1 , p2

( p12 + p22 )s

= C(2, 2m; s) − S(2, 2m; s),

(3.8.1)

where C(n, m; s) =

  cosn mθ p , p 1 2 ( p12 + p22 )s

S(n, m; s) =

  sinn mθ p , p 1 2

. ( p12 + p22 )s (3.8.2) These sums are considered in [16], where it is shown that all elements of the families of sums C(2, 2m; s) and S(2, 2m; s) can be represented as finite combinations of sums from the family C(1, 4m; s). Numerical evidence indicates that the first two families of sums have zeros off the critical line, while the third has its non-trivial zeros on the critical line. Note that, from (3.8.1), C(0, 1; s) = 4L 1 (s)L −4 (s). These remarks are supplemented by two theorems, proofs of which may be found in McPhedran et al. [16]. p1 , p2

,

p1 , p2

Theorem 3.1 Suppose that C(0, 1; s) obeys the Riemann hypothesis. Then C(1, 4m; s) obeys the Riemann hypothesis for any positive integer m. Conversely, if C(1, 4m; s) obeys the Riemann hypothesis for some m then C(0, 1; s)) obeys the Riemann hypothesis. Theorem 3.2 If Theorem 3.1 holds then the distribution functions for the zeros NC1,4 ( 12 , t) and NC0,1 ( 12 , t) must agree in all terms which go to infinity with t. Thus the validity of the Riemann hypothesis for ζ (s) and L −4 (s) together is equivalent to the validity of the Riemann hypothesis for all members of the class of angular sum C(1, 4m; s). Furthermore, the number of zeros of ζ (s) and L −4 (s) together up to any real number t agrees with that for any angular sum C(1, 4m; s) in so far as all terms which diverge with t as it goes to infinity are concerned.

3.9 Commentary: Computational issues of angular lattice sums

149

3.9 Commentary: Computational issues of angular lattice sums (1) The angular lattice sums defined in (3.8.2) can be evaluated numerically from the subset of sums 

p12n

( p1 , p2 )

( p12 + p22 )s+n

C(2n, 1; s) =

.

(3.9.1)

Using the Mellin transform and Hobson’s integral, as described at the end of Section 1.2, we find √ 2 π (s + n − 12 )ζ (2s − 1) C(2n, 1; s) = (s + n)  ∞  ∞  s  p2 s−1/2 8π + (π p1 p2 )n K s+n−1/2 (2π p1 p2 ); (s + n) p1 p1 =1 p2 =1

(3.9.2) here the term involving ζ (2s − 1) comes from the axial contribution (where Hobson’s integral does not converge), in this case when p2 = 0. This sum is exponentially convergent since for large arguments the modified Bessel function of the second kind is approximated by ( π −z e . K n+1/2 (z) ∼ 2z In practice, the sum requires p1 and p2 to be large enough that 2π p1 p2 exceeds |s − 12 | by a factor 2 or so before rapid convergence sets in. All the other angular sums defined in (3.8.2) can be obtained by using trigonometric addition formulae to obtain appropriate combinations of the C(2n, 1; s) (including of course the sums for n = 0). Note also that we have the limit lim C(2n, 1; s) = 2ζ (s).

n→∞

(2) The same technique involving the Mellin transform, Poisson summation, and Hobson’s integral can be used to obtain good approximations for a large class of sums. For instance, we consider S(F, G; s) =

 p0 , p1

F( p1 ) , [G( p1 ) + p02 ]s

where p0 runs over the integers and p1 could stand for a multi-index; the prime excludes those values of p1 , p0 for which the denominator of the summand is zero. We require G to be a non-negative function, increasing as | p1 | → ∞ at least  as rapidly as | p1 |a for some a > 0; moreover, we require that p1 F( p1 )/G( p1 )s

150

Angular lattice sums

converges absolutely for some range of s. Then, we have 1

 2π 2s− 2 ( 12 − s)ζ (1 − 2s) S(F, G; s) = F( p1 ) (s) p1 : G( p1 )=0 √ 1  π (s − 2 ) F( p1 ) + (s) G( p1 )s−1/2 p1 : G( p1 )=0

+

s−1/2  p0 F( p1 ) √ (s) G( p1 ) p0 =1 p1 : G( p1 )=0  % & × K s−1/2 2π p0 G( p1 ) . 4π s

∞ 



(3.9.3)

(3) A useful numerical technique for investigating factorizations of lattice sums is a consideration of their expansions in the direct summation region Re s  1. In this region, the terms of a sum may be readily evaluated in order of powers of increasing distance from the origin. Possible expansions involving say products of Dirichlet L functions may then be tested against the terms arising in the sum in question, using the product rule for the products: ∞ ∞ ∞   ep cn  dm = , s s n m ps n=1

m=1

(3.9.4)

p=1

where on the right-hand side e p is the sum of cn dm over all positive integer pairs (n, m) such that nm = p. The possibility of prefactors of algebraic form as well as products of Dirichlet L-functions has to be taken into account. The knowledge of the form of the functional equation satisfied by the target sum is valuable in assembling appropriate factorization elements. This technique will be useful in obtaining some factorizations in Chapter 4.

3.10 Commentary: Angular lattice sums and the Riemann hypothesis (1) An interesting feature of the lattice sums C(1, 4m; s) is the distribution of gaps between their zeros on the critical line, for m = 0. The distribution functions of the gaps between zeros for C(1, 4; s) and C(0, 1; s) are quite different, particularly for small gaps. The former fits well the Wigner surmise [2, 6] for the Gaussian unitary ensemble:   −4S 2 9S 2 , (3.10.1) P(S) = 2 exp π π where P(S) denotes the probability of finding a gap S in a distribution of gaps whose mean has been scaled to unity. This distribution gives a probability that goes to zero as S 2 for small gaps. By contrast, for C(0, 1; s) [2], the product

3.10 Commentary: Angular lattice sums and the Riemann hypothesis 151 factorization gives an uncorrelated superposition of two unitary ensemble sets, with no clear diminution in the probability as S tends to zero. Thus, it seems possible that the distribution of gaps between zeros might be a good indicator of whether simple factorizations of lattice sums are possible or not. (2) It is of interest to consider non-dimensionalized sums over rectangular lattices in two dimensions of the form [17] σ (0) (d1 , d2 ; s) = (d1 d2 )s =



 ( p12 d12 ( p1 , p2 )=(0,0)

( p12 d1 /d2 p=0)

1 + p22 d22 )s

1 . + p22 d2 /d1 )s

(3.10.2)

This sum may be put into a form suitable for numerical evaluation by the use of Mellin transform techniques, in the usual way. We introduce a general form for sums over Macdonald functions: K (n, m; s; λ) =

 s−1/2 ∞  p2 ( p1 p2 π )n K m+s−1/2 (2π p1 p2 λ). (3.10.3) p1

p1 , p2 =1

The quantity λ is introduced, as usual, as an adjunct in analysis – for example, differentiation with respect to it could generate sums over the derivatives of Macdonald functions and help one to deduce recurrence relations among the sums. It is also useful to introduce two combinations of zeta functions: (s)ζ (2s) (s − 12 )ζ (2s − 1) + , 4π s 4π s−1/2 (s)ζ (2s) (s − 12 )ζ (2s − 1) T− (s) = − . 4π s 4π s−1/2 T+ (s) =

(3.10.4)

The first of these is symmetric under s → 1 − s and the second is antisymmetric. It may be shown that both satisfy the Riemann hypothesis; this was proved for T− (s) by Taylor [27]. We then have σ

(0)

  s−1 √ (s − 12 ) d2 d1 s ζ (2s − 1) (d1 , d2 ; s) = 2ζ (2s) +2 π d2 (s) d1 )   s d1 8π d1 . (3.10.5) + K 0, 0; s; (s) d2 d2 

A similar expansion may be derived by, for example, differentiating with respect to d1 /d2 in the rightmost expression in (3.10.2), while regarding d2 /d1

152

Angular lattice sums

as constant, or following the Mellin transform route: 

p12

p=0)

( p12 d1 /d2 + p22 d2 /d1 )s+1

 s (s + 12 ) √ d2 π ζ (2s − 1) (s + 1) d1 )   d1 8π s d1 . (3.10.6) + K 1, 1; s; (s + 1) d2 d2

=2

(3) The preceding treatment allows us to obtain explicit analytic expressions for Macdonald function sums. We put d1 = d2 = 1 and then have sums which we have already encountered on the left-hand sides of (3.10.5) (see also (3.10.2)) and (3.10.6). From these we get K (0, 0; s; 1) =

(s)ζ (s)L −4 (s) (s)ζ (2s) (s − 12 )ζ (2s − 1) − − (3.10.7) 2π s 4π s 4π s−1/2

and K (1, 1; s; 1) =

1 s(s)ζ (s)L −4 (s) 1 (s − 2 )ζ (2s − 1) − (s − ) . 2 4π s 4π s−1/2

(3.10.8)

K (0, 0; s; 1) is real on the critical line and has zeros both on this line and off it [17] whereas K (1, 1; s; 1) is complex on the critical line. (4) We can find linear combinations of K (0, 0; s; 1) and K (1, 1; s; 1) which are real on the critical line: K (1, 1; s; 1) − 12 s K (0, 0; s; 1) = 14 T+ (s) + 12 (s − 12 )T− (s)

(3.10.9)

and K (1, 1; s; 1) −

1 2 (s

−

1 2 )K (0, 0; s; 1)

(0)   (s)σ2s 1 1 s − T− (s). = + 2 2 32π s (3.10.10)

Both these functions have their zeros on the critical line and close to those of T− (s), since the term (s − 12 ) multiplying this function means that for large t it dominates its partner function. It is easy to furnish a first-order estimate for the shift of the zero away from the corresponding zero of T− (s). (5) Horizontal Riemann curves Fig. 3.4 plots the modulus of the function Z defined by (x, y) → |ζ (x + i y)| for 15 < x < 45 and 1 < y < 100. It is instructive to plot Z (x, y) as a function of x on the critical strip [0, 1] for various values of y; see Fig. 3.5. (Compare what happens when the roles of x and y are reversed.) As it turns out, the monotonicity exhibited in the graph of level curves of Z (·, y) implies the Riemann hypothesis. This conclusion is given in [31]

3.10 Commentary: Angular lattice sums and the Riemann hypothesis 153

Figure 3.4 The modulus of the zeta function.

3 2.5 2 1.5 1 0.5 0 10

15

20

25

30

35

40

t

Figure 3.5 The first few zeros of ζ on the critical line.

and was discovered while the first author of that paper was teaching an undergraduate complex variable class. Much related material and many illustrations are to be found in Conrey’s survey [5]. (6) The Riemann hypothesis as a random walk Recall the Liouville lambda function λ(n) := (−1)r (n) , where r (n) counts the number of prime factors with repetition in n. Fig. 3.6 shows λ plotted as a two-dimensional random walk,

154

Angular lattice sums 0

–200 –400 –600 –800 –1000 –1200 –1400 –1600 –1800 0

200

400

600

800

1000

1200

Figure 3.6 The Riemann hypothesis as a random walk on Liouville’s function.

found by walking with steps (±1, ±1) through pairs of points of the sequence {λ(2k), λ(2k + 1)}nk=1 . The fact that n steps of the walk stay in a box with side √ of length roughly n implies the Riemann hypothesis: that all non-real zeros of the ζ -function have a real part equal to one-half (giving the critical line). The first few zeros are plotted in Fig. 3.6. Note that ∞  λ(n) n=1

ns

=

ζ (2s) . ζ (s)

References [1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York, 1972. [2] E. Bogomolny and P. Leboeuf. Statistical properties of the zeros of zeta functions – beyond the Riemann case. Nonlinearity, 7:1155–1167, 1994. [3] S. K. Chin, N. A. Nicorovici, and R. C. McPhedran. Green’s function and lattice sums for electromagnetic scattering by a square array of cylinders. Phys. Rev. E, 49, 1994. [4] R. Clausius. Die mechanische Behandlung der Electricität. Braunschweig, Vieweg, 1879. [5] J. B. Conrey. The Riemann hypothesis. Not. Amer. Math. Soc., 50:341–353, 2003. [6] B. Dietz and K. Zyczkowski. Level-spacing distributions beyond the Wigner surmise. Z. Phys. Condens. Matter, 84:157–158, 1991.

References

155

[7] J. C. M. Garnett. Colours in metal glasses and in metallic films. Phil. Trans. Roy. Soc. London, 203:385–420, 1904. [8] M. L. Glasser. The evaluation of lattice sums. III. Phase modulated sums. J. Math. Phys., 15:188–189, 1974. [9] M. L. Glasser and I. J. Zucker. Lattice sums. In Theoretical Chemistry, Advances and Perspectives (H. Eyring and D. Henderson, eds.), vol. 5, pp. 67–139, 1980. [10] D. S. Jones. Generalized Functions. McGraw-Hill, New York, 1966. [11] C. Kittel. Introduction to Solid State Physics. Wiley, New York, 1953. [12] R. Landauer. Electrical conductivity in inhomogeneous media. In Proc. Amer. Inst. Phys. Conf., vol. 40, pp. 2–45, 1978. [13] H. A. Lorentz. The Theory of Electrons. B. G. Teubner, Leipzig, 1909. Reprint: Dover, New York, 1952. [14] L. Lorenz. Wiedemannsche Ann., 11:70, 1880. [15] R. C. McPhedran, L. C. Botten, N. P. Nicorovici, and I. J. Zucker. Systematic investigation of two-dimensional static array sums. J. Math. Phys., 48:033501, 2007. [16] R. C. McPhedran, L. C. Botten, D. J. Williamson, and N. A. Nicorovici. The Riemann hypothesis and the zero distribution of angular lattice sums. Proc. Roy. Soc. London A, 467:2462–2478, 2011. [17] R. C. McPhedran, G. H. Smith, N. A. Nicorovici, and L. C. Botten. Distributive and analytic properties of lattice sums. J. Math. Phys., 45:2560–2578, 2004. [18] V. V. Mityushev. Transport properties of doubly-periodic arrays of circular cylinders. Z. Angew. Math. Mech., 77:115–120, 1997. [19] V. V. Mityushev and P. M. Adler. Longitudinal permeability of spatially periodic rectangular arrays of circular cylinders. I. A single cylinder in the unit cell. Z. Angew. Math. Mech., 82:335–345, 2002. [20] O. F. Mossotti. Memorie di Matematica e di Fisica della Società Italiana delle Scienze Residente in Modena, 24:49–74, 1850. [21] A. B. Movchan, N. A. Nicorovici, and R. C. McPhedran. Green’s tensors and lattice sums for elastostatics and elastodynamics. Proc. Roy. Soc. London A, 453:643–662, 1997. [22] N. A. Nicorovici, C. G. Poulton, and R. C. McPhedran. Analytical results for a class of sums involving Bessel functions and square arrays. J. Math. Phys., 37:2043–2052, 1996. [23] W. T. Perrins, D. R. McKenzie, and R. C. McPhedran. Transport properties of regular arrays of cylinders. Proc. Roy. Soc. London A, 369:207–225, 1979. [24] C. G. Poulton, L. C. Botten, R. C. McPhedran, and A. B. Movchan. Source-neutral Green’s functions for periodic problems and their equivalents in electromagnetism. Proc. Roy. Soc. London A, 455:1107–1123, 1999. [25] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev. Integrals and Series. 1. Elementary Functions. Gordon and Breach, New York, 1986. [26] Lord Rayleigh. On the influence of obstacles arranged in rectangular order upon the properties of a medium. Phil. Mag., 34:481–502, 1892. [27] P. R. Taylor. On the Riemann zeta function. Quart. J. Oxford, 16:1–21, 1945. [28] G. N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, 1922.

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[29] E. T. Whittaker and G. N. Watson. A Course of Modern Analysis, 4th edition. Cambridge University Press, Cambridge, 1946. [30] I. J. Zucker. Further relations amongst infinite series and products II. The evaluation of 3-dimensional lattice sums. J. Phys. A, 23:117–132, 1990. [31] P. Zvengrowski and F. Saidak. On the modulus of the Riemann zeta function in the critical strip. Math. Slovaca, 53:145–272, 2003.

4 Use of Dirichlet series with complex characters

4.1 Introduction In Section 1.4, Dirichlet series with real characters were introduced in order to find  2 2 −s closed forms for lattice sums of the type ∞ m,n=0,0 (am + bmn + cn ) . Here, Dirichlet series with complex characters and their application to finding closed forms for other types of lattice sums will be discussed. We first recapitulate the properties of real L-series given in Chapter 1 with additional definitions of certain properties and some new notation. This will enable a straightforward comparison to be made with the complex L-series to be considered immediately after that.

4.2 Properties of L-series with real characters The properties of L-series with real characters are well known and have been given elsewhere; see e.g., Ayoub [2], Zucker and Robertson [17]. Elementary L-series (modulo k) are given by L k (s) =

∞  χ k (n) n=1

ns

;

(4.2.1)

k will be referred to as the period or order of the L-series. The quantity χ k is called a character; it is a multiplicative function and periodic in k and is defined as follows. For integers m, n, χ k (1) = 1,

χ k (n + k) = χ k (n), χ k (n) = 0

χ k (mn) = χ k (m)χ k (n),

if gcd(k, n) = 1.

(4.2.2)

It was shown by Dickson [6] that χ k can only assume values which are the φ(k)th roots of unity, where φ(k) is the Euler totient function giving the number of positive integers less than k which are relatively prime to k. Thus characters may be real or complex.

158

Use of Dirichlet series with complex characters

For real L-series χ k (n) can only be ±1. All real L-series then divide into two types, according to whether χ k (k − 1) = ±1. If χ k (k − 1) = +1 then the series will be said to have positive parity. For such series the signs of the characters will be mirrored exactly in the midpoint of the series. If χ k (k −1) = −1 then the signs of the characters will be mirrored by opposite signs. Theorems given in Ayoub [2] show that the only possible characters are given by the Legendre–Jacobi– Kronecker symbol: χ k (n) = (n|k). The number of independent real-character L-series is then found to be as follows. Let P = 1 or let P be a product of all different primes, i.e., odd and square-free; then (1) (2) (3) (4)

if k = P there is just one primitive L-series, if k = 4P there is just one primitive L-series, if k = 8P there are two primitive L-series, if k = 2P, 2α P, where α > 3 or P is not square free, there is no primitive L-series.

The definition of a primitive L-series is complicated. A good account is given by Ayoub [2], and it is discussed fully in Zucker and Robertson [17]. The parity of a real L-series is determined as follows. (1) (2) (3) (4) (5)

If k If k If k If k If k

= P ≡ 1 (mod 4) the L-series has positive parity. = P ≡ 3 (mod 4) the L-series has negative parity. = 4P with P ≡ 3 (mod 4) the L-series has positive parity. = 4P with P ≡ 1 (mod 4) the L-series has negative parity. = 8P there is an L-series of each parity.

The suffix k will be signed according to whether χ k (k − 1) = ±1. If s is a positive integer then explicit formulae for L −k (2s − 1) and L k (2s) may be established. They are k (1 − n/k) (−1)s−1 22s−2 π 2s−1  B χ −k (n) 2s−1 L −k (2s − 1) = √ (2s − 1)! k n=1

(4.2.3)

and L k (2s) =

k (−1)s−1 22s−1 π 2s  B (1 − n/k) χ k (n) 2s , √ (2s)! k n=1

(4.2.4)

where Bs (x) are the Bernoulli polynomials. As both n and k are positive integers and Bs (1 − n/k) are rational numbers then, for s a positive integer, √ L −k (2s − 1) = R(k) k π 2s−1 , √ L k (2s) = R  (k) k π 2s ,

4.2 Properties of L-series with real characters

159

where R(k) and R  (k) are rational numbers. It is also known that 2h(k) L k (1) = √ log 0 , k where h(k) is the class number of the binary quadratic√form of discriminant k and 0 is the fundamental unit of the number field Q( k). Some examples of real-character L-series are given below. It is pertinent to introduce here another notation for L-series, which emphasizes their periodicity. Let (k, l : s) := (k, l) =

∞  n=0

1 ; (kn + l)s

(4.2.5)

then L k (s) =

k−1 

χ k (l)(k, l).

(4.2.6)

l=1

Note that (k, l : s) may be expressed as ∞ 1  1 1  l = ζ s, , ks (n + l/k)s ks k

(4.2.7)

n=0

where ζ (s, q) denotes the Hurwitz zeta function. (For numerical and symbolic computations of (k, l), see the commentary in Chapter 8.) The Riemann zeta function is thus expressed as follows:   1 1 1 (4.2.8) ζ (s) = L 1 (s) = 1 + s + s + s + · · · = (1, 1). 2 3 4 For the Catalan (or Dirichlet) β function we have   1 1 1 β(s) = L −4 (s) = 1 − s + s − s + · · · = (4, 1) − (4, 3). 3 5 7

(4.2.9)

Other real-character L-series which will be used here are L −3 (s) = (3, 1) − (3, 2), L 5 (s) = (5, 1) − (5, 2) − (5, 3) + (5, 4), L −7 (s) = (7, 1) + (7, 2) − (7, 3) + (7, 4) − (7, 5) − (7, 6), L −8 (s) = (8, 1) + (8, 3) − (8, 5) − (8, 7), L 8 (s) = (8, 1) − (8, 3) − (8, 5) + (8, 7), L 12 (s) = (12, 1) − (12, 5) − (12, 7) + (12, 11), L −20 (s) = (20, 1) + (20, 3) + (20, 7) + (20, 9) − (20, 11) − (20, 13) − (20, 17) − (20, 19), L 28 (s) = (28, 1) + (28, 3) − (28, 5) + (28, 9) − (28, 11) − (28, 13) − (28, 15) − (28, 17) + (28, 19) − (28, 23) + (28, 25) + (28, 27).

160

Use of Dirichlet series with complex characters

In addition to these there is always one further real L-series of period k formed  by all the characters χ k (n) equal to 1; this can be shown to be equal to (1 − pn−s )L 1 (s), where the pn are all the different prime factors of k. This is illustrated below for k = 5 by making use of the expansion property of (k, l); thus (k, l) = (2k, l) + (2k, k + l) = (3k, l) + (3k, k + l) + (3k, 2k + l) = · · · . So (1, 1) = (5, 1) + (5, 2) + (5, 3) + (5, 4) + (5, 5), but (5, 5) = 5−s (1, 1) and so (5, 1) + (5, 2) + (5, 3) + (5, 4) = (1, 1) − (5, 5) = (1 − 5−s )L 1 . As explained in Chapter 1, these real L-series are used to express double sums given by Q(a, b, c; s) = Q(a, b, c) =

∞ 

(am 2 + bmn + cn 2 )−s

(4.2.10)

m,n=0,0

as sums of products of pairs of real L-series of opposite parity. (Please note the change of notation from S used in Chapter 1 to Q as used here. The symbol S will be required for another type of sum later on). Some examples are Q(1, 0, 5) = L 1 L −20 + L −4 L 5 , Q(1, 1, 2) = 2L 1 L −7 ,   2Q(5, 2, 10) = 1 + 71−2s L 1 L −4 − L −7 L 28 . In particular, many Q(1, 0, λ) have been solved by expressing them as the Mellin transforms of the product of the Jacobian θ3 (q) functions with different arguments. Thus  ∞ ∞    1 s−1 Q(1, 0, λ) = t exp −m 2 t − λn 2 t dt (s) 0 m,n=0,0  ∞

1 = t s−1 θ3 (q)θ3 (q λ ) − 1 dt, where e−t = q (s) 0 and θ3 (q) =

∞ 

2

q n = 1 + 2q 2 + 2q 4 + 2q 9 + · · · .

n=−∞

By expressing θ3 (q)θ3 (q λ ) − 1 as a Lambert series where possible, the integral is easily evaluated in terms of L-series. A long list of such results may be found in Table 1.6 and supplemented by others in Zucker and Robertson [18]. Not all Q can be thus factored, and the possible use of complex L-series has long been contemplated by Zucker as possibly providing a vehicle for exposing non-susceptible Q to possible factorization. However, it turns out that complex L-series are useful

4.3 Properties of L-series with complex characters

161

in factorizing another kind of two-dimensional lattice sum, so an account of these will now be given.

4.3 Properties of L-series with complex characters In order to compare real L-series with complex L-series, we first list all the possible L-series with both real and complex characters for periods k = 1−10 and for k = 16. For k a prime, these are found by taking each φ(k)th root of unity in turn and using the rules for characters to produce a given series, always starting with +1 for the first term. If k has factors then these factors also have period k and will reduce the number of independent L-series of period k. This will become evident from the following. For For

k = 1, k = 2,

φ(1) = 1; φ(2) = 1;

(1, 1) = L 1 . (2, 1) = (1 − 2−s )L 1 .

For k = 3, φ(3) = 2 as 1 and 2 are relatively prime to 3. There are two possible order-3 L-series: (3, 1) + (3, 2) = (1 − 3−s )L 1 ,

(3, 1) − (3, 2) = L −3 .

For k = 4, φ(4) = 2 as 1 and 3 are relatively prime to 4. There are two possible order-4 L-series: (4, 1) + (4, 3) = (1 − 2−s )L 1 ,

(4, 1) − (4, 3) = L −4 .

For k = 5, φ(5) = 4 and the four roots of unity are ±1 and ±i, giving four order-5 L-series: (5, 1) + (5, 2) + (5, 3) + (5, 4) = (1 − 5−s )L 1 , (5, 1) − (5, 2) − (5, 3) + (5, 4) = L 5 , (5, 1) + i(5, 2) − i(5, 3) − (5, 4) = L i−5 , (5, 1) − i(5, 2) + i(5, 3) − (5, 4) = L −i −5 . For k = 6, φ(6) = 2, as 1 and 5 are relatively prime to 6. There are just two possible L-series, both real and of lower order than 6: (6, 1) + (6, 5) = (1 − 2−s )(1 − 3−s )L 1 , (6, 1) − (6, 5) = (1 + 2−s )L −3 . For k = 7, φ(7) = 6 as 1, 2, 3, 4, 5, 6 are relatively prime to 7. The sixth roots of unity are 1, −1, ω, −ω, ω2 , −ω2 where ω = exp(iπ/3). The six possible L-series are

162

Use of Dirichlet series with complex characters (7, 1) + (7, 2) + (7, 3) + (7, 4) + (7, 5) + (7, 6) = (1 − 7−s )L 1 , (7, 1) + (7, 2) − (7, 3) + (7, 4) − (7, 5) − (7, 6) = L −7 ,

(7, 1) + ω2 (7, 2) + ω(7, 3) − ω(7, 4) − ω2 (7, 5) − (7, 6) = L ω−7 , 2

(7, 1) − ω(7, 2) − ω2 (7, 3) + ω2 (7, 4) + ω(7, 5) − (7, 6) = L −ω −7 , (7, 1) + ω2 (7, 2) − ω(7, 3) − ω(7, 4) + ω2 (7, 5) + (7, 6) = L ω7 , 2

(7, 1) − ω(7, 2) + ω2 (7, 3) + ω2 (7, 4) − ω(7, 5) + (7, 6) = L −ω 7 . For k = 8, φ(8) = 4 as 1, 3, 5, 7 are relatively prime to 8. The four order-8 L-series are all real: (8, 1) + (8, 3) + (8, 5) + (8, 7) = (1 − 2−s )L 1 , (8, 1) − (8, 3) + (8, 5) − (8, 7) = L −4 , (8, 1) + (8, 3) − (8, 5) − (8, 7) = L −8 , (8, 1) − (8, 3) − (8, 5) + (8, 7) = L 8 . For k = 9, φ(9) = 6 as 1, 2, 4, 5, 7, 8 are relatively prime to 9. There are six order-9 L-series: (9, 1) + (9, 2) + (9, 4) + (9, 5) + (9, 7) + (9, 8) = (1 − 3−s )L 1 , (9, 1) − (9, 2) + (9, 4) − (9, 5) + (9, 7) − (9, 8) = L −3 , (9, 1) − w (9, 2) − w(9, 4) + w(9, 5)] + w2 (9, 7) − (9, 8) = L −ω −9 , 2

2

(9, 1) + w(9, 2) + w2 (9, 4) − w 2 (9, 5) − w(9, 7) − (9, 8) = L ω−9 ,

(9, 1) + w2 (9, 2) − w(9, 4) − w(9, 5) + w2 (9, 7) + (9, 8) = L ω9 , 2

(9, 1) − w(9, 2) + w2 (9, 4) + w 2 (9, 5) − w(9, 7) + (9, 8) = L −ω 9 . For k = 10, φ(10) = 4 as 1, 3, 7, 9 are relatively prime to 10. The four possible L-series are all of lower order than 10: (10, 1) + (10, 3) + (10, 7) + (10, 9) = (1 − 2−s )(1 − 5−s )L 1 , (10, 1) − (10, 3) − (10, 7) + (10, 9) = (1 + 2−s )L 5 , (10, 1) − i(10, 3) + i(10, 7) − (10, 9) = (1 − i2−s )L i−5 , (10, 1) + i(10, 3) − i(10, 7) − (10, 9) = (1 + i2−s )L −i −5 . For k = 16, φ(16) = 8 as 1, 3, 5, 7, 9, 11, 13, 15 are relatively prime to 16. So, there are eight L-series. As φ(k) gets larger the depiction of L-series in (k, l) symbols can become unwieldy. It is thus convenient here to introduce a slightly modified notation, namely a signed (k, l) symbol defined by (k, l)+ := (k, l) + (k, k − l),

(k, l)− := (k, l) − (k, k − l).

(4.3.1)

4.3 Properties of L-series with complex characters

163

This enables us to halve the number of symbols required to show an L-series. It will be seen that positive-parity series are described entirely by (k, l)+ symbols and negative-parity series by (k, l)− terms. Thus for k = 16 we have (16, 1)+ + (16, 3)+ + (16, 5)+ + (16, 7)+ = (1 − 2−s )L 1 , (16, 1)− − (16, 3)− + (16, 5)− − (16, 7)− = L −4 , (16, 1)− + (16, 3)− − (16, 5)− − (16, 7)− = L −8 , (16, 1)+ − (16, 3)+ − (16, 5)+ + (16, 7)+ = L 8 , (16, 1)− + i(16, 3)− + i(16, 5)− + (16, 7)− = L i−16 , (16, 1)− − i(16, 3)− − i(16, 5)− + (16, 7)− = L −i −16 , (16, 1)+ + i(16, 3)+ − i(16, 5)+ − (16, 7)+ = L i16 , (16, 1)+ − i(16, 3)+ + i(16, 5)+ − (16, 7)+ = L −i 16 . These results illustrate most properties that have been observed for complex L-series and allow us to make the following conjectures regarding them. (1) The number of positive-parity series always equals the number of negativeparity series. (2) The number of complex-character positive-parity L-series is even and they divide into pairs of complex conjugates. The same is true for complexcharacter L-series with negative parity. (3) The first and the last terms of all L-series are always real. (4) Knowing the parity of the series and the first non-real term then, using the properties of characters given in (4.2.2), the coefficients of all the other terms may be established. This allows us to create a concise notation for complex L-series. The subscript gives the parity and period whilst the superscript gives the coefficient of the first non-real term, and this specifies the given series completely. (5) The number of (k, l) symbols needed to specify a given L-series equals the number of L-series for a given k. Thus every (k, l) is expressible as a linear combination of L-series of period k. We note from the above that for k = 6, 10 the inclusion of complex L-series does not yield any L-series with these periods, and this has also been found for k = 14 and 18. So, part of the statement made in Section 4.2 regarding real L-series seems to hold for complex L-series, namely, if k is equal to twice an odd number then no L-series of such a period exists. It would appear that this is a result of the fact that φ(2n) = φ(n) if n is an odd number. However, whereas, for real L-series, if k is a perfect square > 4 then no L-series of such period are found, it is seen here that, for such k, complex L-series of that period do exist.

164

Use of Dirichlet series with complex characters

It is necessary to point out here that all the statements about real L-series made in the previous section have been proved whereas the conjectures made in this section about complex L-series have not been proved.

4.4 Expressions for displaced lattice sums in closed form for j = 2−10 Displaced lattice sums were introduced by McPhedran et al. [11, 12]. They are of the form    −s   r 2 p 2 + n+ (4.4.1) S( p, r, j; s) := S( p, r, j) = m+ j j m,n and are related to the Green’s function connected with sums over the square lattice. There are also associated phased lattice sums,  exp[2πi(mp/j + nr/j)] , (m 2 + n 2 )s

σ ( p, r, j; s) = σ ( p, r, j) =

(4.4.2)

m,n=0,0

the displaced and phased lattice sums being connected by the Poisson summation formula. It is easily seen that S( p, r, j) = S( j − p, j −r, j), so all the independent S for a given j will be found by allowing both p and r to take on all values up to and including j/2. Then the displaced lattice sums may be expressed as the following Mellin transform: S( p, r, j) =

 m,n

=



j 2s (s)

m + p/j

 0

∞

2

1

 + n + r/j



j 2s

2 2 s m,n ( jm + p) + ( jn + r )

,  exp − ( jm + p)2 t + ( jn + r )2 t dt.

∞ 

t s−1

2 s =

m,n=−∞

(4.4.3) The exponential sums are disjoint and each can be evaluated separately. Then, letting e−t = q and writing ∞ 

q ( jm+ p) = θ ( j, p), 2

m=−∞

we have S( p, r, j) =

j 2s (s)

 0

∞

t s−1 θ ( j, p)θ ( j, r ) dt.

(4.4.4)

4.4 Expressions for displaced lattice sums in closed form for j = 2−10 165 For small j, θ ( j, p) may be expressed in terms of θ3 -functions, see [15], and we list the results for j = 2, 3, 4, 6:   % 2& θ (2, 1) = θ3 (q) − θ3 q 4 , θ ( j, 0) = θ3 q j ,    1 1 θ3 (q) − θ3 q 9 , θ3 (q) − θ3 (q 4 ) , θ (4, 1) = θ (3, 1) = 2 2   4 16 , θ (6, 0) = θ3 (q 36 ), θ (4, 2) = θ3 (q ) − θ3 q  1 θ3 (q) − θ3 (q 4 ) − θ3 (q 9 ) + θ3 (q 36 ) , θ (6, 1) = 2  1 4 θ3 (q ) − θ3 (q 36 ) , θ (6, 2) = θ (6, 3) = θ3 (q 9 ) − θ3 (q 36 ). (4.4.5) 2 For these j values all the independent S( p, r, j) may be expressed in terms of Q(1, 0, λ), using  ∞ ∞ 

1 t s−1 exp −(m 2 + λn 2 )t dt, Q(1, 0, λ) = (s) 0 m,n=0,0  ∞

1 = t s−1 θ3 (q)θ3 (q λ ) − 1 dt, where e−t = q. (4.4.6) (s) 0 Thus we obtain

S(0, 1, 2) = 22s Q(1, 0, 4) − 4−s Q(1, 0, 1) ,

S(1, 1, 2) = 22s Q(1, 0, 1) − 2Q(1, 0, 4) + 4−s Q(1, 0, 1) , S(0, 1, 3) = S(1, 1, 3) = S(0, 1, 4) = S(0, 2, 4) = S(1, 1, 4) = S(1, 2, 4) =

32s 2 32s 2 42s 2 42s 2 42s 2 42s 2



Q(1, 0, 9) − 9−s Q(1, 0, 1) ,



Q(1, 0, 1) − 2Q(1, 0, 9) + 9−s Q(1, 0, 1) ,



Q(1, 0, 16) − 4−s Q(1, 0, 4) ,

  2−2s Q(1, 0, 4) − 2−4s Q(1, 0, 1) ,

Q(1, 0, 1) − 2Q(1, 0, 4) + 4−s Q(1, 0, 4) ,



Q(1, 0, 4) − 4−s Q(1, 0, 1) − Q(1, 0, 16) − 4−s Q(1, 0, 4) , (4.4.7)

S(0, 0, 6) = Q(1, 0, 1), 1  −2s 6 Q(1, 0, 1) − 3−2s Q(1, 0, 4) − 2−2s Q(1, 0, 9) S(0, 1, 6) = 2 + Q(1, 0, 36)] ,

166

Use of Dirichlet series with complex characters 62s (1 + 2−2s + 3−2s + 6−2s )Q(1, 0, 1) 4 − 2(1 + 3−2s )Q(1, 0, 4) − 2(1 + 2−2s )Q(1, 0, 9)

+ 2Q(4, 0, 9) + 2Q(1, 0, 36) ,  32s  −2s −3 Q(1, 0, 1) + Q(1, 0, 9) , S(0, 2, 6) = 2 62s −2s −2 (1 + 3−2s )Q(1, 0, 1) + (1 + 3−2s )Q(1, 0, 4) S(1, 2, 6) = 4

+ 21−2s Q(1, 0, 9) − Q(4, 0, 9) − Q(1, 0, 36) ,  32s  (1 + 3−2s )Q(1, 0, 1) − 2Q(1, 0, 9) , S(2, 2, 6) = 2 S(0, 3, 6) = −Q(1, 0, 1) + 2−2s Q(1, 0, 4),

S(1, 1, 6) =

S(1, 3, 6) =

62s (−3−2s − 6−2s )Q(1, 0, 1) + 2 × 3−2s Q(1, 0, 4) 2

+ (1 + 2−2s )Q(1, 0, 9) − Q(4, 0, 9) − Q(1, 0, 36) ,

62s −2s 6 Q(1, 0, 1) − 3−2s Q(1, 0, 4) − 2−2s Q(1, 0, 9) 2

+ Q(4, 0, 9) ,

S(3, 3, 6) = 22s (1 + 2−2s )Q(1, 0, 1) − 2Q(1, 0, 4) . (4.4.8) S2, 3, 6) =

Of the six different Q(a, b, c) which appear in the preceding displaced sums, the values of four have been found in terms of Dirichlet series with real characters. They are Q(1, 0, 1) = 4L 1 (s)L −4 (s),

Q(1, 0, 4) = 2(1 − 2−s + 21−2s )L 1 (s)L −4 (s),

Q(1, 0, 9) = (1 + 31−2s )L 1 (s)L −4 (s) + L −3 (s)L 12 (s), Q(1, 0, 16) = (1 − 2−s + 21−2s − 21−3s + 22−4s )L 1 (s)L −4 (s) + L −8 (s)L 8 (s). The functions Q(1, 0, 36) and Q(4, 0, 9) cannot be individually found (for Q(1, 0, 36) see also the commentary in Section 4.6). However, it will be seen that ± [Q(1, 0, 36) + Q(4, 0, 9)] appear together in three of the cases considered and it may be shown, via the theory of which numbers can be represented by the binary quadratic forms (m 2 + 36n 2 ) and (4m 2 + 9n 2 ), that1 Q(1, 0, 36) + Q(4, 0, 9) = (1 − 2−s + 21−2s )(1 + 31−2s )L 1 (s)L −4 (s) + (1 + 2−s + 21−2s )L 3 (s)L 12 (s). So, we are able to express eight of the ten possible S( p, r, 6) in terms of known Dirichlet series. All the results for j = 2 − 4 and j = 6 are listed below. 1 We are grateful to Mark Watkins, University of Bristol, for obtaining this result for us.

4.4 Expressions for displaced lattice sums in closed form for j = 2−10 167 For j = 2, S(0, 1, 2) = 22s+1 (1 − 2−s )L 1 (s)L −4 (s), S(1, 1, 2) = 2s+2 (1 − 2−s )L 1 (s)L −4 (s). For j = 3, 32s 2 32s S(1, 1, 3) = 2

S(0, 1, 3) =

  (1 − 3−2s )L 1 (s)L −4 (s) + L −3 (s)L 12 (s) ,   (1 − 3−2s )L 1 (s)L −4 (s) − L −3 (s)L 12 (s) .

For j = 4,

42s (1 − 2−s )L 1 (s)L −4 (s) + L −8 (s)L 8 (s) , 2 S(0, 2, 4) = 22s+1 (1 − 2−s )L 1 (s)L −4 (s),

S(0, 1, 4) =

S(1, 1, 4) = 23s (1 − 2−s )L 1 (s)L −4 (s), S(1, 2, 4) =

42s (1 − 2−s )L 1 (s)L −4 (s) − L −8 (s)L 8 (s) . 2

For j = 6, 1 (−3 + 21+s − 21+2s − 32s )L 1 (s)L −4 (s) 2

− 32s L −3 (s)L 12 (s) + 62s Q(1, 0, 36) ,  1 s (2 − 1)(32s − 1)L 1 (s)L −4 (s) + 32s (2s + 1)L −3 (s)L 12 (s) , S(1, 1, 6) = 2  1  2s (3 − 1)L 1 (s)L −4 (s) + 32s L −3 (s)L 12 (s) , S(0, 2, 6) = 2  1 s s 2 (2 + 1)(32s − 1)L 1 (s)L −4 (s) − 18s (2s + 1)L −3 (s)L 12 (s) , S(1, 2, 6) = 4  1  2s (3 − 1)L 1 (s)L −4 (s) − 32s L −3 (s)L 12 (s) , S(2, 2, 6) = 2 S(0, 3, 6) = 21+s (2s − 1)L 1 (s)L −4 (s),  1 s (2 − 1)(32s − 1)L 1 (s)L −4 (s) − 32s (2s + 1)L −3 (s)L 12 (s) , S(1, 3, 6) = 2 1 S(2, 3, 6) = (−3 + 21+s − 21+2s − 32s )L 1 (s)L −4 (s) 2

− 32s L −3 (s)L 12 (s) + 62s Q(4, 0, 9) ,

S(0, 1, 6) =

S(3, 3, 6) = 4(2s − 1)L 1 (s)L −4 (s).

168

Use of Dirichlet series with complex characters

It may also be seen that the sum of two unknown members may also be expressed in terms of Dirichlet series; thus S(0, 1, 6) + S(2, 3, 6) = 2s−1 (2s − 1)(32s − 1)L 1 (s)L −4 (s) + 2s−1 32s (2s + 1)L −3 (s)L 12 (s). Whether Q(1, 0, 36) or Q(4, 0, 9) may be individually expressed in terms of Dirichlet series with complex characters is still an open question. For j = 5, the previous method can be used to establish one factorization, see [15]: S(1, 2, 5) = (5s − 1)L 1 (s)L −4 (s). The other results for j = 5 all involve Dirichlet series with complex characters and have been obtained, see McPhedran et al. [11], by evaluating coefficients ∞ −s for a sufficient set of c . These cn in the expansion S(k, l, 5) = n n=1 cn n may be compared with the corresponding expansions generated from appropriate combinations of Dirichlet functions, where, if a product with a pair of complex characters occurs, this must be accompanied by the product with complex conjugated characters, to ensure a real result for Im s = 0. The following two complex L-functions of order 20 were required: L i20 (s) = (20, 1)+ + i(20, 3)+ − i(20, 7)+ − (20, 9)+ , L −i 20 (s) = (20, 1)+ − i(20, 3)+ + i(20, 7)+ − (20, 9)+ . Then, defining a5(s) = (1 − 5−s )2 L 1 (s)L −4 (s) − L 5 (s)L −20 (s), b5(s) = (1 − 5−s )2 L 1 (s)L −4 (s) + L 5 (s)L −20 (s), −i c51(s) = L i−5 (s)L i20 (s) + L −i −5 (s)L 20 (s),   −i (s)L (s) , c52(s) = i L i−5 (s)L i20 (s) − L −i −5 20

we have the solutions 52s [b5(s) + c51(s)], 4 52s S(1, 1, 5) = [a5(s) − c52(s)], 4 52s S(0, 2, 5) = [b5(s) − c51(s)], 4 52s S(2, 2, 5) = [a5(s) + c52(s)]. 4 S(0, 1, 5) =

Note that, in the expansion of S( p, r, j) as a sum over factors 1/n s , all terms with non-zero coefficients have n ≡ ( p 2 + r 2 ) mod j. For j = 5, each modulus

4.4 Expressions for displaced lattice sums in closed form for j = 2−10 169 value 1, 2, 3, 4, or 5 occurs only once. We believe the results given above are the first examples of the expression of real lattice sums in terms of real and complex L-series. Alternating forms of these sums can also be found; for example, (−1)m+n 1

s = (1 − 5−s )2 (1 − 21−s )L 1 (s)L −4 (s) 2 2 4 m,n=−∞ (5m) + (5n + 1) ∞ 

+ (1 + 21−s )L 5 (s)L −20 (s) + (1 − i21−s )L i20 (s)L i−5 (s)

 −i + (1 + i21−s )L −i (s)L (s) . 20 −5

(4.4.9)

1−s )L (s) → log 2. It is simple The two sides converge for s = 1. As s → 1, (1−2 1  √ √ to show that L −4 (1) = π/4, L 5 (1) = 2 log ( 5 + 1)/2 / 5, and L −20 (1) = √ π/ 5. The evaluation of L i−5 (1) and L i20 (1) is somewhat more difficult. We find √   2 π 3 + 4i 1/4 L i−5 (1) = , 5 5

and, with the aid of Mathematica, L i20 (1)

' '   √ √ √ 1 =− √ 5 − 5 log −1 + 5 − 5 − 2 5 5 2 ' '   √ √ √ + 5 + 5 log 1 + 5 − 5 + 2 5 ' '   √ √ √ − i 5 + 5 log −1 + 5 + 5 − 2 5 ' '   √ √ √ + 5 − 5 log 1 + 5 − 5 + 2 5 .

−i Similar results for L −i −5 (1) and L 20 (1) lead to  ∞  √ (−1)m+n π log(11 + 5 5) = 2 2 25 (5m) + (5n + 1) m,n=−∞ '  √ √ √  − 5 log 5 + 1 − 5 + 2 5 . (4.4.10)

In the same way, the following results may also be obtained. √ ∞  (11 + 5 5) (−1)m+n π log = 25 4 (5m + 1)2 + (5n + 1)2 m,n=−∞ √ '   √ √ 5π + log 5−1+ 5−2 5 , 25

170

Use of Dirichlet series with complex characters ∞ 

√ (−1)m+n π log(11 + 5 = − 5) 25 (5m)2 + (5n + 2)2 m,n=−∞ √ '   √ √ 5π log − 5+1− 5+2 5 , 25 √ ∞ m+n  (−1) π (11 + 5 5) = log 25 4 (5m + 2)2 + (5n + 2)2 m,n=−∞ √ '   √ √ 5π − log 5−1+ 5−2 5 , 25 √ ∞ m+n  (−1) 5+1 π = log 2 − log 2 2 5 2 m,n=−∞ m + (5n + 1) √ '   √ √ 5 log −19+9 5+3 85−38 5 , + 5 and ∞ 

(−1)m+n

π =− 2 2 5 m,n=−∞ m + (5n + 2)



√

log 2 − log

5+1 2

'   √ √ 5 − log −19 + 9 5 + 3 85 − 38 5 . 5 √

This illustrates that, since the sum is real, combinations of complex L-series series will lead to a real number, which in this case could be expressed in elementary constants of analysis. Equation (4.4.10) was submitted as a problem to the American Mathematical Monthly by Zucker and McPhedran [16]. It stimulated Berndt et al. [3] to seek alternative methods to solve similar problems. They considered F(a,b) (x) :=

∞ 

(−1)m+n , (xm)2 + (an + b)2 m,n=−∞

and showed it was possible to evaluate the latter for any positive rational value of x and for many values of (a, b) ∈ N2 . They proved that F(a,b) (x) = −

a−1 2π  −(2 j−1)b ω ax j=0 ∞

 2 j+1 2m+1 −2 j−1 2m+1 log (1 − ω q )(1 − ω q ) , m=0

where

ω = eπi/a

and

q = e−π/x .

(4.4.11)

4.4 Expressions for displaced lattice sums in closed form for j = 2−10 171 Then, using elliptic functions, singular moduli, class invariants, and the Rogers– Ramanujan continued fraction, they solved (4.4.11) for a ∈ {3, 4, 5, 6} and for a variety of examples, producing such specimens as  √  (−1)m+n π log 8(4 − = 15) , √ m 2 + 5(3n + 1)2 9 5 m,n=−∞  1/8  ∞  2 +1 (−1)m+n π , = √ log 1/8 (8m)2 + (4n + 1)2 2 −1 8 2 m,n=−∞ ∞ 

as well as the problem sum (4.4.11). For larger even values of a another formula for F(a,b) (x) in terms of the Jacobian elliptic function dn was given, and an evaluation of F10,1 (1) was carried out. The methods of Berndt et al. [3] seem to be completely disjoint from the procedures employed here. Following the method described above for j = 5 it has been possible to find similar results for larger values of j. For j = 7, there are nine independent lattice sums. Of these three on their own have been solved. Pairs of the other six sums could also be evaluated in terms of real and complex L-series. New complex L-series of order 28 were required, and these were (28, 1)− + ω(28, 3)− + ω2 (28, 5)− + ω2 (28, 9)− + ω(28, 11)− + (28, 13)− = L ω−28 , (28, 1)− − ω2 (28, 3)− − ω(28, 5)− − ω(28, 9)− − ω2 (28, 11)− + (28, 13)− = L −ω −28 , 2

(28, 1)+ − ω(28, 3)+ − ω2 (28, 5)+ + ω2 (28, 9)+ + ω(28, 11)+ − (28, 13)+ = L −ω 28 , (28, 1)+ + ω2 (28, 3)+ + ω(28, 5)+ − ω(28, 9)+ − ω2 (28, 11)+ − (28, 13)+ = L ω28 . 2

We let a7(s) = (1 − 7−2s )L 1 (s)L −4 (s) − L −7 (s)L 28 (s), b7(s) = (1 − 7−2s )L 1 (s)L −4 (s) + L −7 (s)L 28 (s), −ω c71(s) = L ω7 (s)L ω−28 (s) + L −ω 7 (s)L −28 (s), 2

2

−ω ω c72(s) = L ω−7 (s)L −ω 28 (s) + L −7 (s)L −28 (s), 2

2

−ω c73(s) = ωL ω7 (s)L ω−28 (s) − ω2 L −ω 7 (s)L −28 (s), 2

2

2 −ω ω c74(s) = ωL ω−7 (s)L −ω 28 (s) − ω L −7 (s)L −28 (s), 2

2

ω2 (s) ω −ω2 L −28 (s) − ωL −ω 7 (s)L −28 (s),

c75(s) = ω2 L 7

−ω ω c76(s) = ω2 L ω−7 (s)L −ω 28 (s) − ωL −7 (s)L −28 (s). 2

2

172

Use of Dirichlet series with complex characters

Then 72s [a7(s) + c71(s) − c72(s)] , 12 72s [a7(s) − c73(s) + c74(s)] , S(1, 2, 7) = 12 72s [a7(s) + c75(s) − c76(s)] . S(1, 3, 7) = 12

S(2, 3, 7) =

These three sums, which have been solved, have expansions involving terms with values 6, 5, and 3 (mod 7) respectively: 72s [b7(s) + c71(s) + c72(s)] , 6 72s [b7(s) − c73(s) − c74(s)] , S(0, 3, 7) + S(1, 1, 7) = 6 72s [b7(s) + c75(s) + c76(s)] . S(0, 2, 7) + S(3, 3, 7) = 6 S(0, 1, 7) + S(2, 2, 7) =

The three sets of pair relations above contain terms with values 1, 2, and 4 (mod 7) respectively. For j = 8, let a8(s) = (1 − 2−s )L 1 (s)L −4 (s) − L −8 (s)L 8 (s), b8(s) = (1 − 2−s )L 1 (s)L −4 (s) + L −8 (s)L 8 (s), −i i i c81(s) = L −i −16 (s)L 16 (s) + L −16 (s)L 16 (s),   −i i i (s)L (s) − L (s)L (s) . c82(s) = i L −i 16 −16 −16 16

The following three individual sums have been found: S(1, 2, 8) = 26s−3 [a8(s) + c82(s)] , S(1, 3, 8) = 25s−2 a8(s), S(2, 3, 8) = 26s−3 [a8(s) − c82(s)] . The six remaining independent sums occur as three pairs: S(0, 1, 8) + S(1, 4, 8) = 26s−2 [b8(s) + c81(s)] , S(0, 3, 8) + S(3, 4, 8) = 26s−2 [b8(s) − c81(s)] , S(1, 1, 8) + S(3, 3, 8) = 25s−1 b8(s).

4.4 Expressions for displaced lattice sums in closed form for j = 2−10 173 For j = 9 there are 12 independent terms; the following L-series of order 36 were required: (36, 1)− − ω(36, 5)− − ω2 (36, 7)− − ω2 (36, 11)− − ω(36, 13)− + (36, 17)− = L −ω −36 , (36, 1)− + ω2 (36, 5)− + ω(36, 7)− + ω(36, 11)− + ω2 (36, 13)− + (36, 17)− = L ω−36 , 2

(36, 1)+ + ω(36, 5)− − ω2 (36, 7)+ + ω2 (36, 11)+ − ω(36, 13)+ − (36, 17)+ = L ω36 , (36, 1)+ − ω2 (36, 5)− + ω(36, 7)+ − ω(36, 11)+ + ω2 (36, 13)+ − (36, 17)+ = L −ω 36 . 2

Let a9(s) = (1 − 3−2s )L 1 (s)L −4 (s) − L −3 (s)L 12 (s), b9(s) = (1 − 3−2s )L 1 (s)L −4 (s) + L −3 (s)L 12 (s), −ω ω c91(s) = L ω9 (s)L −ω −36 (s) + L 9 (s)L −36 (s), 2

2

−ω ω ω c92(s) = L −ω −9 (s)L 36 (s) + L −9 (s)L 36 (s), 2

2

2 −ω ω c93(s) = ωL ω9 (s)L −ω −36 (s) − ω L 9 (s)L −36 (s), 2

2

−ω ω 2 ω c94(s) = ωL −ω −9 (s)L 36 (s) − ω L −9 (s)L 36 (s), 2

2

−ω ω c95(s) = ω2 L ω9 (s)L −ω −36 (s) − ωL 9 (s)L −36 (s), 2

2

−ω ω ω c96(s) = ω2 L −ω −9 (s)L 36 (s) − ωL −9 (s)L 36 (s). 2

2

The 12 sums divide into six pairs, giving S(2, 2, 9) + 2S(1, 4, 9) = S(1, 1, 9) + 2S(2, 4, 9) = S(4, 4, 9) + 2S(1, 2, 9) = S(0, 1, 9) + 2S(1, 3, 9) = S(0, 4, 9) + 2S(3, 4, 9) = S(0, 2, 9) + 2S(2, 3, 9) =

92s 6 92s 6 92s 6 92s 6 92s 6 92s 6

[a9(s) + c91(s) − c92(s)] , [a9(s) − c93(s) + c94(s)] , [a9(s) + c95(s) − c96(s)] , [b9(s) + c91(s) + c92(s)] , [b9(s) − c93(s) − c94(s)] , [b9(s) + c95(s) + c96(s)] .

174

Use of Dirichlet series with complex characters

These results as displayed are associated with terms having the values 8, 2, 5, 1, 7, 4 (mod 9) respectively. The similarity between these results and those for j = 7 is noteworthy. For j = 10 there are 13 independent terms and the complex L-series of order 20, given for j = 5, were required. Let a10(s) = (1 − 2−s )(1 − 5−s )2 L 1 (s)L −4 (s) − (1 + 2−s )L 5 (s)L −20 (s), b10(s) = (1 − 2−s )(1 − 5−s )2 L 1 (s)L −4 (s) + (1 + 2−s )L 5 (s)L −20 (s), −i c10(s) = L i−5 (s)L i20 (s) + L −i −5 (s)L 20 (s),   −i d10(s) = i L i−5 (s)L i20 (s) − L −i −5 (s)L 20 (s) .

Seven of the 13 independent terms could be solved exactly:

102s b10(s) + c10(s) − 2−s d10(s) , s+2 2 S(1, 3, 10) = 10s (1 − 2−s )(1 − 5−s )L 1 (s)L −4 (s), S(1, 1, 10) =

S(1, 4, 10) = S(1, 5, 10) = S(2, 3, 10) = S(3, 3, 10) = S(3, 5, 10) =

102s a10(s) − 2−s c10(s) − d10(s) , 8

102s −s a10(s) + 2 c10(s) + d10(s) , 2s+2

102s a10(s) + 2−s c10(s) + d10(s) , 8

102s b10(s) − c10(s) + 2−s d10(s) , s+2 2

102s a10(s) − 2−s c10(s) − d10(s) . 2s+2

The remaining six form three pairs which have solutions:

102s b10(s) + c10(s) − 2−s d10(s) , 4

102s S(0, 3, 10) + S(2, 5, 10) = b10(s) − c10(s) + 2−s d10(s) , 4 102s S(1, 2, 10) + S(3, 4, 10) = s (1 − 2−s )(1 − 5−s )L 1 (s)L −4 (s). 5

S(0, 1, 10) + S(4, 5, 10) =

It appears to us that there is no apparent reason why similar results cannot be found for larger values of j. However, at this stage we have no rules for determining which particular S( p, r, j) or combination of such terms can be put into closed form, and some criteria are desirable in order to go further.

4.5 Exact solutions of lattice sums involving indefinite quadratic forms 175

4.5 Exact solutions of lattice sums involving indefinite quadratic forms Efforts to solve Q(a, b, c) (see (4.2.10)) in terms of L-series have concentrated on the case when the binary quadratic form am 2 +bmn +cn 2 is positive definite, i.e., a > 0 and the discriminant b2 − 4ac < 0. Following Lorenz [9], who found an exact form for T (1, 0−1), defined below, Zucker and Robertson [18] attempted to solve certain lattice sums involving indefinite quadratic forms. They investigated  | p 2 m 2 − r 2 n 2 |−s = T ( p 2 , 0, −r 2 ; s) := T ( p 2 , 0, −r 2 ), (4.5.1) p 2 m 2 =r 2 n 2

and in particular found an expression for T (1, 0, −r 2 ) in terms of (k, l) symbols. This was r −1   

2 (2r, t) + (2r, 2r − t) . T (1, 0, −r 2 ) = 4r −2s 1 − 21−s + 21−2s L 21 + 2 t=1

(4.5.2) For r = 1−6 it was possible to find solutions in terms of the squares of positiveparity real L-series, but for larger values of r no such solutions could be found. Now, it was noted in Section 4.3 that every (k, l) is expressible as a linear combination of L-series of period k if the complex L-series are included. Thus in principle (4.5.2) may be written in L-series for every r . To illustrate this, closed forms for r = 1 − 13 have been evaluated; the outcome is displayed in Table 4.1. In this table we have  1 T (1, 0, −r 2 ) = 2 − r 2 n 2 |s |m 2 2 2 m =r n

and

 ω = exp

iπ 3



 ,

τ = exp

iπ 5



 ,

ρ = exp

iπ 6

 .

These results show that in addition to squares of positive-parity L-series with real characters, products of pairs of positive-parity complex L-series will in general be required in order to solve T (1, 0, −r 2 ). It is apparent that with sufficient labour there is no limit on how far one may go, but until now no clear pattern has yet emerged. We have established numerically, but not analytically, the following functional equation:  sπ   π 2s−1 (1 − s) ? tan . T (1, 0, −r 2 ; s) = T (1, 0, −r 2 ; 1 − s) r (s) 2 It may also be shown that T has the following expansion near its second-order pole at s = 1:  2 2 2 4γ + γ + 2γ1 + C(r ) + O[s]. T (1, 0, −r 2 ; 1 + s) = 2 + rs r rs

176

Use of Dirichlet series with complex characters Table 4.1 The functions T (1, 0, −r 2 ) for r = 1−13

r 1 2 3 4 5 6

7 8

9

10

11

12

13

T (1, 0, −r 2 )   4 1 − 21−s + 21−2s L 21 (s)  √    √  2 1+ 2 − 1 2−s + 21−2s 1 − 2 + 1 2−s + 21−2s L 21 (s)    2 1 − 21−s + 21−2s 1 − 2 × 3−s + 31−2s L 21 (s)    1 − 2−s + 21−2s 1 − 2−s − 21−3s + 22−4s L 21 (s) + L 28 (s)      1 − 21−s + 21−2s 1 − 2 × 5−s + 51−2s L 21 (s) + 1 + 21s + 21−2s L 25 (s)  √    √  1+ 2 − 1 2−s + 21−2s 1 − 2 + 1 2s + 21−2s   × 1 − 2 × 3−s + 31−2s L 21 (s) + L 212 (s)      2 1 − 21−s + 21−2s ω2 1 − 2 × 7−s + 71−2s L 21 (s) + 43 1 + 2s + 21−2s L −ω 7 (s)L 7 (s) 3   1 1 − 21−s +3 × 2−2s − 22−3s +3 × 21−4s − 23−5s + 3 × 22−6s − 24−7s + 24−8s L 2 (s) 1 2

+ 12 (1 + 21−2s )L 28 (s) + L i16 (s)L −i 16 (s)   √  √    2 1 − 21−s + 21−2s 1+ 3 − 1 3−s + 31−2s 1 − 3 + 1 3−s + 31−2s L 21 (s) 3   ω2 + 43 1 + 2−s + 21−2s L −ω 9 (s)L 9 (s)  √  √     1 1+ 2 − 1 2−s + 21−2s 1 − 2 + 1 2−s + 21−2s 1 − 2 × 5−s + 51−2s L 21 (s) 2   √  √    + 12 1 + − 2 + 1 2−s + 21−2s 1 + 2 + 1 2−s + 21−2s L 25 (s) + L i20 (s)L −i 20 (s)      √ −1−s 2 1 − 21−s + 21−2s 4 2 −s 1−2s + 21−2s 1 − 2 × 11 + 11 L 1 (s) + 5 1 + (1 − 5)2 5   √ −1−s 3 2 4 τ4 + 21−2s L −τ × L τ11 (s)L −τ 11 (s)L 11 (s) 11 (s) + 5 1 + (1 + 5)2     1 1 − 2 × 3−s + 31−2s 1 − 2−s + 21−2s 1 − 2s − 21−3s + 22−4s L 21 (s) 2     + 12 1 + 2 × 3−s + 31−2s L 28 (s) + 12 1 + 21−2s L 212 (s) + 12 L 224 (s)      1 1 − 21−s + 21−2s 1 − 2 × 13−s + 131−2s L 21 (s) + 13 1 + 2−1−s + 21−2s L 213 (s) 3   2   ρ −ρ 4 −ρ 2 ρ4 + 23 1 − 2−s + 21−2s L 13 (s)L 13 (s) + 23 1 + 2s + 21−2s L 13 (s)L 13 (s)

The quantity γ1 is the first Stieltjes constant (which occurs in the Laurent series expansion of the Hurwitz zeta function), and C(r ) is a constant depending on the integer r . Its first three values are C(1) = 2 log2 2,

C(2) =

3 log2 2, 2

C(3) =

2 1 log2 2 + log2 3. 3 3

The expansion near s = 0 is then T (1, 0, −r 2 ; s) = 1 + 2 log

π  r

s + O(s 2 ).

4.6 Commentary: Quadratic forms and closed forms

177

In this section it has been shown that the introduction of L-series with complex characters has enabled closed forms to be established for many twodimensional lattice sums which previously could not be expressed in this fashion. An obvious question is whether complex-character L-series can play a role in higher-dimensional sums, and this may be an interesting path to pursue.

4.6 Commentary: Quadratic forms and closed forms (1) The collaboration of M. L. Glasser and I. J. Zucker on lattice sums commenced with their mutual interest in finding closed forms for double sums of the form Q(a, b, c; s) =

∞ 

(am 2 + bmn + cn 2 )−s ,

m,n=0,0

where a > 0 and its discriminant b2 − 4ac < 0. In this case the binary quadratic form is positive definite. In Chapter 1 we discussed how, for these forms which had one class per genus, the double sum above could be expressed as sums of pairs of real-character L-series of opposite parity, and the results are displayed in Table 1.6. Altogether, 101 of these discriminants are known, and if any others exist it is known there can be at most only one more. At one time it was thought that these were the only Q sums expressible in this manner. However, it was later found, in certain cases of discriminants with two classes per genus, that if the two classes which appeared in one genus were the related quadratic forms am 2 ± bmn + cn 2 then, since Q(a, b, c) = Q(a, −b, c), the lattice sum involving these quadratic forms could also be found as sums of pairs of real L-series. Several examples were given in [18]. These turned out to be in the nature of flukes. A theorem of Chowla [5] shows that the number of two-class-per-genus discriminants must also be finite. (2) In this chapter the successful use of L-series with complex characters in expressing some two-dimensional displaced lattice sums in closed form has been demonstrated. Further, the particular set of lattice sums involving the indefinite quadratic forms T (1, 0, −r 2 ) has also been given in closed form using both real and complex L-series. This, as mentioned in the main part of this chapter, has raised the hope that the latter may be applicable to other sums involving positive definite quadratic forms. Immediate candidates for analysis would be Q(1, 0, 36) and Q(4, 0, 9), whose sum in terms real L-series is known as is also the sum of the displaced lattice sums S(0, 1, 6) and S(2, 3, 6). Since all other displaced lattice sums of the form S( p, q, 6) could individually be expressed in terms of real L-series, the exclusion of S(0, 1, 6) and S(2, 3, 6) seems an anomaly. There are no period-3 or period-6 L-series with complex characters. So if a solution is possible then it seems likely that period9 complex L-series would be involved. But whether they should be combined

178

Use of Dirichlet series with complex characters

with period-16 or period-36 complex L-series is not clear. Also, algebraic factors such as (1 − 2−s + 21−2s ) and (1 + 31−2s ) may complicate matters even further, so as yet no definite statement can be made about Q(1, 0, 36) and Q(4, 0, 9) individually. The reader should also consult Chapter 8, where lattice sums involving arbitrary quadratic forms are analyzed. (3) In a recent work by Guillera and Rogers [8], it has been shown that certain sums of the form ∞  n!3 a − bn −n z (4.6.1) 1 (s)n ( 2 )n (1 − s)n n 3 n=1

may be evaluated as a Dirichlet L-series at 2. The summand of (4.6.1) is essentially the reciprocal of the summand of a Ramanujan series for 1/π and, as such, sums of this form are called companion series by the authors. When a companion series is divergent, the value (4.6.1) is given by the analytic continuation of its hypergeometric representation. We now very briefly outline the methods used in [8]. The idea of completing a hypergeometric function is introduced, that is, replacing the summation index n in a Ramanujan series by n + x, and extending the summation range to Z. The result is a function  (s)n+x ( 1 )n+x (1 − s)n+x 2 z n+x , Yx (z) = 3 (1) n+x n∈Z which is clearly periodic in x with period 1. This fact, combined with complex analytic arguments – including an estimate of Yx (z) when Im x is large, allow the authors to relate Yx (z) to the original series by some trigonometric terms. The companion series is then extracted as a coefficient in the x-expansion of Yx (z). The coefficients are extracted, after significant work aided by θ -function identities, as integrals of Eisenstein series. Finally, using trigonometric partial fractions, the companion series is expressed as the sum of two terms, each of the form Q(a, b, c; s). Hence, when both Q’s are solvable in terms of L-series (using Table 1.6), the companion series (4.6.1) also simplifies in terms of L-series. However, when one Q does not (apparently) reduce to L-series, taking a suitable hypergeometric lefthand side of the companion series allows one to rapidly compute the lattice sum. In particular, [8] shows that Q(1, 0, 36; 2) evaluates to a linear combination of L −4 (2), L −3 (2), and a 5 F4 with argument ≈ −4 × 10−7 , giving very fast convergence. Similar expressions for Q(1, 0, 20; 2) and Q(1, 0, 52; 2) are also given.

4.7 Commentary: More on numerical discovery Commentaries 3.9 and 3.10 at the end of Chapter 3 describe how one can discover a simple factorization of a lattice sum as a product of L-series, by comparing terms

4.8 Commentary: A Gaussian integer zeta function

179

 in the series expansion n cn /n s of both sides, or by matching the distribution of zeros on the critical line. We remark that, numerically, it appears that all L-series, whether real or complex, of the same period k have the same distribution of zeros on the critical line; namely, the number of zeros on 1/2 + it with t ∈ [0, N ] is asymptotically   N Nk log −1 . 2π 2π Here, we describe another method that can be used to discover a factorization. As a toy example, consider equation (4.4.9). The left-hand side sum may be easily evaluated naïvely to (say) 25 digits for s = 4 (often a suitable integral can give many more digits). If we do not know the right-hand side, we could reasonably guess that (for fixed s) it will be a linear combination of (1 − 5−s )2 L 1 (s)L −4 (s),

L 5 (s)L −20 (s),

L i20 (s)L i−5 (s),

−i L −i 20 (s)L −5 (s),

based on the factorization of S(a, b, 5) derived earlier in the chapter. (In fact, knowing that both sides must converge when s = 1, we could argue that the factor 1 − 21−s must also appear in front of the L 1 (S) term.) Now, each of the four products of L-series can be easily computed to high precision, and we input these values, as well as the approximate sum, as a vector v into an integer relation program such as PSLQ (see e.g., [4]). The PSLQ algorithm, which is well implemented in Maple, takes an input vector v of real or complex numbers and attempts to find a vector u of elements in R, where R is usually the integers (but can also be the Gaussian integers), such that v · u = 0 within the prescribed precision. In this example, even setting the precision to as low as 20 digits (we can afford to do this because our guess has only four terms) returns, for s = 4, u = {7, 9, 8 − i, 8 + i, −32}, from which we readily guess that 7 = 2s−1 − 1, 9 = 2s−1 + 1, 32 = 2s+1 , etc., and therefore recover (4.4.9).

4.8 Commentary: A Gaussian integer zeta function One may evaluate in closed form  1  1 ζG (N ) := = zN (m + in) N m,n Z (i)

for positive even-integer N > 1. Here, as always, the primes denote that summation avoids the pole at 0. Note that ζG (2n) has been introduced already as S2n in Chapter 3, equation (3.1.7). The sum ζG (2n) is also a special case of an Eisenstein series.

180

Use of Dirichlet series with complex characters

Hint: The evaluation is implicitly covered in [4, pp. 167–70] and in [10]. It relies on analysis of the Weierstrass ℘-function; see [14, Chapter 23]. We recall that ℘ (x) :=

 m,n

1 1 − . 2 (2in + 2im − x) (2in + 2im)2

We then differentiate twice and extract the coefficients of ζG (2n). For N divisible by 4, the sum is actually is a rational multiple of powers of the Weierstrass invariants     1 4 1 1 4 β = g2 = K √ 4 4 2 and g3 = 0. This is the so-called lemniscate case of ℘. Here β(x) := B(x, x) = (x)2 / (2x) is the central beta function. One can show that the general formula for N ≥ 1 is ζG (4N ) = where p1 = 1/20 and

  4N 1 pN 2K √ , 4N − 1 2

 N −1 pm p N −m 3 m=1 , pN = (4N + 1)(2N − 3)

for N > 1; compare with (3.2.5). The next three values are p2 = 1/1200, p3 = 1/156000 and p4 = 1/21216000. The corresponding values of q N := 16 N p N are 16 128 256 4 , , , . 5 75 4875 82875 Finally note that q N satisfies the same recursion as PN . This leads to the simpler expression qN K ζG (4N ) = 4N − 1



1 √ 2

4N

qN = 4N − 1



β( 14 ) 4

4N .

(4.8.1)

By contrast, it is easy to show that ζG (2N + 1) = ζG (4N + 2) = 0. Note that ζG is the two-dimensions analogue of the Riemann zeta function for even arguments, since, for N > 1,   1 0, N odd, = N n 2ζ (N ), N even. n

4.9 Commentary: Gaussian quadrature

181

Moreover, we have the closed form ζ (2N ) =

&2N (−1) N +1 b2N (−1) N +1 b2N % (2π )2N = 2β( 12 ) , 2(2N )! 2(2N )!

(4.8.2)

where b N is the N th Bernoulli number, defined by the exponential generating  N t function ∞ N =0 b N t /N ! = t/(e − 1). The generalized quantum sum is defined by  1 , Y(a, b, c) := (z − a)(z − b)(z − c) m,n=0

where z = m + ni and a, b, c are not on the lattice. One may show that Y has a formal power series development in ζG (n), by writing the summand as     a a2 1 b b2 c c2 1 + + + + + · · · 1 + + · · · 1 + + · · · . z z z z3 z2 z2 z2 This leads, for |a|, |b|, |c| < 1, to Y(a, b, c) =

∞ 

π4n−3 (a, b, c) ζG (4n),

(4.8.3)

n=1

where πn (a, b, c) :=



a k1 b k2 c k3 .

k1 ,k2 ,k3 ≥0 k1 +k2 +k3 =n

In particular, πn (a, a, a) = 12 (n+1)(n+2) a n . There is a corresponding expansion for more than three linear terms.

4.9 Commentary: Gaussian quadrature The classical theory of Gaussian quadrature is a scheme for approximating integrals by finite sums involving orthogonal polynomials (see [1, Chapter 5]). Here we describe a new, curious, and not yet fully explored procedure to numerically evaluate lattice sums using Gaussian quadrature. We recall some facts about orthogonal polynomials. For a reasonably wellbehaved function ω(x), there exists a sequence of polynomials { pn (x)}∞ n=0 (where n is also the degree), such that they are orthogonal with respect to ω:  b pm (x) pn (x) ω(x) dx = h n δmn . (4.9.1) a

Here δmn is the Kronecker delta. Every polynomial of degree n can be expressed uniquely as a linear combination of p0 , . . . , pn . We may normalize pn so that they

182

Use of Dirichlet series with complex characters

are monic, in which case it is easy to show that they satisfy the recurrence relation pn+1 = (x − an ) pn (x) − bn pn−1 (x),

with bn = h n / h n−1 . (4.9.2) √ 2 (For example, taking −a = b = ∞, ω(x) = e−x /2 , h n = 2π n!, an = 0, and bn = n we get the Hermite polynomials.) With this set-up, we have Proposition 4.1 (Gaussian quadrature) For a reasonable function f ,  b n  f (x) ω(x) d x = f (xi )wi + Rn , (4.9.3) a

i=1

where xi are the roots of pn (x), wi are weights, defined by wi =

−h n , pn+1 (xi ) pn (xi )

(4.9.4)

and Rn is the error, which depends on the (2n)th derivative of f . In particular, if f is a polynomial of degree ≤ 2n − 1, Rn = 0 and the quadrature is exact. By its very construction using the roots of the polynomials pn , Gaussian quadrature is exact for polynomials of degree up to n − 1; orthogonality gives exactness for higher degrees and it is this pleasant property which makes Gaussian quadrature superior to many other schemes. In practice, the error is difficult to compute and is often best estimated on a case-by-case basis by fixing f and increasing n. Heuristically, if f is closely approximated by polynomials on (a, b) (e.g., if it has a close-fitting Taylor series) then Gaussian quadrature tends to work well – thus we need to choose ω carefully to make f well behaved. Engblom [7] was one of the first to give a comprehensive report on Gaussian quadrature using discrete measures. He noted that the classical theory carried over  exactly if ω is discrete, that is, if x pm (x) pn (x)ω(x) = h n δmn ; then  x

f (x)ω(x) ≈

n 

f (xi )wi .

(4.9.5)

i=1

(An example of such a class of polynomials is the Poisson–Charlier polynomials, where ω(x) = a x /x!.) Monien [13], using reciprocal polynomials, considered and found applications for orthogonal conditions of the form  1 11 qm 2 qn 2 2 = h n δmn . x x x x>0

He also found the recurrence and closed form (in terms of Bessel functions) for qn (x). Quadrature using qn works particularly well if the summand f (x) admits an asymptotic expansion in powers of 1/x 2 for large x. Here the classical theory still applies: one computes the roots xi of qn (x), from which wi follows.

4.9 Commentary: Gaussian quadrature

183

Since qn (1/x 2 ) is used instead of qn (x), the right-hand side of (4.9.5) becomes n √ i=1 f (1/ x i )wi . Given some ω, it remains to compute xi and wi for quadrature to work. Note that for fixed n we only need to calculate these numbers once – to several hundred significant figures say, so, once set up, the cost for approximating a range of sums is very small. First, we generate a table of orthogonal polynomials pn using the Gram– Schmidt orthonormalization process. We then find the roots, using for example Newton’s method (when the table is small, e.g., less than 50 polynomials) or exploiting the fact that the n simple zeros of pn are eigenvalues of a tridiagonal matrix, and stable and fast algorithms exist for finding them. The weights can then be computed, either using (4.9.4) or by noting that Gaussian quadrature is exact for low-degree elementary polynomials times ω (whose sums may be found independently). In this way, finding wi is equivalent to inverting a Vandermonde matrix whose columns are powers of xi . We now demonstrate how Gaussian quadrature can be applied to lattice sums. As a very simple example, we look at the Hardy–Lorenz sum  1 = 4β(s)ζ (s). (4.9.6) 2 + n 2 )s (m m,n Consider the s = 1 case (the argument is similar for other integers s). We can in fact perform one summation exactly, since ∞  n=−∞

m2

1 π = coth(mπ ), 2 m +n

and we differentiate both sides with respect to m. The resulting sum (replacing m by x) behaves like π/(4x 3 ) − 1/(2x 4 ) for large x, so the polynomials f n with orthogonality conditions ∞  x=1

fn

1 x

fm

1 1 = δmn h n x x2

(4.9.7)

can be used for quadrature. Using the Gram–Schmidt process, we obtain f 0 (x) = 1, f 1 (x) = x − 6ζ (3)/π 2 , . . . The roots of f n (x) serve as xi , the weights are computed using (4.9.4), and the quadrature rule becomes ∞  x=1

f (x)

n 1  1 wi . ≈ f xi x2 i=1

Using n = 20 (i.e., we only need to compute f 20 together with the roots and weights), we obtain 31 correct digits for the s = 2 case of the Hardy–Lorenz sum. Of course, for many lattice sums it is not possible to perform one summation explicitly; also, the sums may be alternating. Our next example tackles both these

184

Use of Dirichlet series with complex characters

problems. Consider S=

 (−1)m+n . (m 2 + n 2 )2 m,n

For Gaussian quadrature, we use the polynomials gn with orthogonality conditions   1   1  (−1)x gm = δmn h n . gn (4.9.8) x x x x Gram–Schmidt gives g0 (x) = 1, g1 (x) = x − π 2 /(12 log 2), . . . We compute xi and wi as before and use the quadrature rule ∞  x=1

f (x)

n 1  (−1)x wi . f ≈ x xi i=1

We perform a double quadrature on S, that is, we first apply quadrature to f (n) = mn/(m 2 + n 2 )2 (since ω(x) = (−1)x /x). This gives a finite sum over functions of m, upon which we perform quadrature as our new function. Using only n = 10, this method gives 15 correct digits for S. Double Gaussian quadrature using f n and gn works well for sums of the type  m,n

(±1)m (±1)n % , am 2 + bmn + cn 2 + dm + en + f )s

so it can be applied to S( p, r, j; s), Q(a, b, c; s), and F(a,b) (x), studied in this chapter, especially when other numerical methods, for instance Mellin transforms and θ -functions, are not straightforward to apply. We normally need to split the sum up naturally into 1 ≤ m, n ≤ ∞ and a few other similar regions. In practice, convergence sets in even for low n though quadrature seems to be slowest when the sum is not alternating. Finally, note that we are not restricted to double sums. As just one example, using triple Gaussian quadrature, the polynomials gn , and n = 25, 36 significant figures for the Madelung constant have been obtained by J. Wan (unpublished, 2012).

References [1] G. E. Andrews, R. Askey, and R. Roy. Special Functions. Cambridge University Press, Cambridge, 1999. [2] R. Ayoub. An Introduction to the Analytic Theory of Numbers. American Mathematical Society, Providence, RI, 1963. [3] B. C. Berndt, G. Lamb, and M. Rogers. Two dimensional series evaluations via the elliptic functions of Ramanujan and Jacobi. Ramanujan Journal, 2011.

References

185

[4] J. M. Borwein and D. H. Bailey. Mathematics by Experiment: Plausible Reasoning in the 21st Century, 2nd edition. A. K. Peters, 2008. [5] S. Chowla. An extension of Heilbronn’s class number theorem. Quart. J. Math. Oxford, 5:304–307, 1934. [6] L. E. Dickson. An Introduction to the Theory of Numbers. Dover, New York, 1957. [7] S. Engblom. Gaussian quadratures with respect to discrete measures. Uppsala University Technical Reports, 7:1–17, 2006. [8] J. Guillera and M. Rogers. Ramanujan series upside-down. Preprint. [9] L. Lorenz. Bidrag tiltalienes theori. Tidsskrift Math., 1:97–114, 1871. [10] P. McKean and V. Moll. Elliptic Curves: Function Theory, Geometry, Arithmetic. Cambridge University Press, New York, 1997. [11] R. C. McPhedran, L. C. Botten, N. P. Nicorovici, and I. J. Zucker. Systematic investigation of two-dimensional static array sums. J. Math. Phys., 48:033501, 2007. [12] R. C. McPhedran, L. C. Botten, N. P. Nicorovici, and I. J. Zucker. On the Riemann property of angular lattice sums and the one-dimensional limit of two-dimensional lattice sums. Proc. Roy. Soc. A, 464:3327–3352, 2008. [13] H. Monien. Gaussian quadrature for sums: a rapidly convergent summation scheme. Math. Comp., 79:857–869, 2010. [14] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark (eds.). NIST Handbook of Mathematical Functions. Cambridge University Press, New York, 2010. [15] I. J. Zucker. Further relations amongst infinite series and products II. The evaluation of 3-dimensional lattice sums. J. Phys. A, 23:117–132, 1990. [16] I. J. Zucker and R. C. McPhedran. Problem 11294. Amer. Math. Monthly, 114:452, 2007. [17] I. J. Zucker and M. M. Robertson. Some properties of Dirichlet L-series. J. Phys. A, 9:1207–1214, 1976. [18] I. J. Zucker and M. M. Robertson. Further aspects of the evaluation of  2 2 −s (m,n=0,0) (am + bmn + cn ) . Math. Proc. Camb. Phil. Soc., 95:5–13, 1984.

5 Lattice sums and Ramanujan’s modular equations

A modular equation of order n is essentially some algebraic relation between theta functions of arguments q and q n respectively. In his notebooks Ramanujan gave many such relations involving Lambert series, and Berndt [1] painstakingly collected these results, proved or verified them, and if necessary made corrections. However, these relations as given by Ramanujan appear in a haphazard way, and there seems to be no systematic way of ordering them or for that matter knowing whether the formulae are independent or complete. Here an attempt is made to arrange these results in a systematic fashion. It will also be demonstrated how new modular relations may be derived from those previously established. Indeed it will be shown how, using sign and Poisson transformations to be described in Chapter 6, each modular equation is essentially a set of either four or eight relations which can be generated from any one of the set by a group of simple transformations. Further it will also be shown how the use of character notation provides a shorthand for the lengthy Lambert series involved. A connection between binary quadratic forms and modular equations will be demonstrated. This will allow very simple proofs of certain modular and mixed modular equations to be executed. Finally it will be exhibited how the Mellin transforms of modular equations, in which q is replaced by e−t , lead to the evaluation of lattice sums. All the modular equations are given in terms of the theta functions described in Section 1.3, namely θ2 =

∞ 

q

(n−1/2)2

,

θ3 =

−∞

∞ 

n2

q ,

θ4 =

−∞

θ5 = 2

∞ 

−∞

(−1)n q (2n−1/2) .

−∞

∞ 

2

2

(−1)n q n ,

Lattice sums and Ramanujan’s modular equations

187

Ramanujan has his own notation, and the following ‘dictionary’ applies: Ramanujan classical

φ(q) θ3

ψ(q 2 ) θ2 /2q 1/4

φ(−q) θ4

ψ(−q 2 ) θ5 /2q 1/4

The main aspects of character notation will now be described. Characters were introduced in Section 1.4 in the definition of Dirichlet L-series. Only real characters are considered here. Primitive characters are defined as follows. If p is a prime > 2 then χ p (n) = χ p = (n| p), where (n| p) is the Legendre–Jacobi–Kronecker symbol, which is ±1. The suffix p will be signed according to whether χ p ( p − 1) = ±1. The simplest possible character is χ 1 (n) = 1. Other examples are as follows: χ −3 (n) = +1,

n ≡ 1 (mod 3),

= −1,

n ≡ 2 (mod 3),

= 0,

n ≡ 0 (mod 3),

χ 5 (n) = +1,

n ≡ 1, 4 (mod 5),

= −1,

n ≡ 2, 3 (mod 5),

= 0,

n ≡ 0 (mod 5).

In addition to these there are three other primitive characters. They are χ −4 = +1,

n ≡ 1 (mod 4),

= −1,

n ≡ 3 (mod 4),

= 0,

(n, 4) = 1,

and χ −8 (n) = +1,

n ≡ 1, 3 (mod 8),

= −1,

n ≡ 5, 7 (mod 8),

= 0,

(n, 8) = 1,

χ 8 (n) = +1,

n ≡ 1, 7 (mod 8),

= −1,

n ≡ 3, 5 (mod 8),

= 0,

(n, 8) = 1.

Further primitive characters may be formed from square-free products of prime characters, and from prime characters multiplied by χ −4 , χ −8 , and χ 8 . Thus χ P = χ p1 χ p2 χ p3 · · · and χ −4 χ P , χ −8 χ P , and χ 8 χ P , are all primitive characters. The character series L P = L P (s) =

∞  χ P (n) n=1

ns

188

Lattice sums and Ramanujan’s modular equations

each yield an independent Dirichlet L-function. For example, L 1 (s) =

∞  χ 1 (n) n=1

ns

= 1−s + 2−s + 3−s + · · · = ζ (s) (Riemann zeta function),

L −3 (s) =

∞  χ −3 (n) n=1

ns

= 1−s − 2−s + 4−s − 5−s + · · · .

(5.0.1) (5.0.2)

Two other types of character are encountered. These are as follows: χ ka (n) = +1

if (k, n) = 1,

= 0

if (k, n) = 1;

χ kb (n) = +1 = −(k − 1)

if (k, n) = 1, if n ≡ 0 (mod k).

(5.0.3)

(5.0.4)

These latter non-primitive characters are essentially modifications of the principal character, χ 1 (n) = +1 for all n. When these non-primitive characters multiply primitive characters the latter are modified in such a way that the character series produced are L-series of the primitive character, multiplied by a factor. Thus it may be shown that ∞  χ P χ ka n=1 ∞  n=1

ns χ P χ kb ns

= 1 − (k|P)k −s L P ,

(5.0.5)

  = 1 − (k|P)k 1−s L P .

(5.0.6)

Of these non-primitive characters, two occur often in this combination, namely χ 2a and χ 2b , and when they multiply primitive characters the combination may be represented by χ 2Pa and χ 2Pb . Two such modified characters appear often here in connection with modular equations of order 3. They are χ −3 χ 2a = χ −6a and χ −3 χ 2b = χ −6b , whose character tables are χ −6a (n) = +1,

n ≡ 1 (mod 6),

= −1,

n ≡ 5 (mod 6),

= 0, χ −6b (n) = +1,

n ≡ 0 (mod 3), n ≡ 1, 2 (mod 6),

= −1,

n ≡ 4, 5 (mod 6),

= 0,

n ≡ 0 (mod 3).

Lattice sums and Ramanujan’s modular equations

189

Then it is found that ∞  χ −6a n=1

ns

−s

= (1 + 2 )L −3 ,

∞  χ −6b n=1

ns

= (1 + 21−s )L −3 .

The values of characters for the χ ±n which appear here have been given, for all n up to 30, in Table A.2 in the Appendix. The period of the character is denoted by a vertical bar, | . A modular equation will be said to have dimension d if the power of the theta function term is d. Thus expressions such as θ (q)θ (q 3 ) or θ 3 (q)/θ (q 3 ) both have dimension 2. The reason for describing such expressions in this way is, as will be seen later, that the Mellin transforms of these forms lead to multiple sums over d-dimensional lattices. Ramanujan’s modular equations are generally given in the form of a relation between theta functions and Lambert series, where Lambert series have the general form ∞  f (n)q mn+k , 1 ± q r (mn+k) n=1

with m, k, and r all integers. To illustrate the use of character notation and the various transformations possible, two pairs of modular equations of order 3, one pair of dimension 2 and the other of dimension 4 will be considered. They are the pair of equations numbered (3.i) and (3.ii) and the pair (3.iii) and (3.iv) in Chapter 19 – referred to henceforth as C.19 – of Ramanujan’s Notebooks Part III [1]:   q q5 q7 q 11 − + − + · · · , 4qψ(q 2 )ψ(q 6 ) = 4 1 − q2 1 − q 10 1 − q 14 1 − q 22 (5.0.7)   q2 q q4 q5 − + − + · · · φ(q)φ(q 3 ) = 1 + 2 1−q 1 + q2 1 + q4 1 − q5 (5.0.8) and q 2q 2 4q 4 5q 5 + + + + ··· , (5.0.9) 1 − q2 1 − q4 1 − q8 1 − q 10   4q 4 q 5q 5 7q 7 + φ 2 (q)φ 2 (q 3 ) = 1 + 4 + + + · · · . 1−q 1 − q7 1 − q4 1 − q5 (5.0.10)

qψ 2 (q)ψ 2 (q 3 ) =

Consider the first pair. It is immediately apparent that these expressions are concisely expressed in character notation, and replacing Ramanujan’s functions with the classical notation we have

190

Lattice sums and Ramanujan’s modular equations θ2 θ2 (q ) = 4 3

∞  χ −6a q n n=1

θ3 θ3 (q 3 ) = 1 + 2

,

1 − q 2n

(5.0.11)

∞ 

χ −3 q n

n=1

1 + (−q)n

.

(5.0.12)

Now if we apply the sign transform to these expressions, a further two equations may be obtained. It is simple to do this to (5.0.8), with the following result: θ4 θ4 (q 3 ) − 1 = −2

∞  χ −6b q n n=1

1 + qn

,

(5.0.13)

which is exactly what would be obtained by a Poisson transform of (5.0.7). The sign transform of (5.0.7) requires care as q has to be replaced by iq. This yields ∞  χ 12 q n q n − q 5n = 4 . 1 + q 6n 1 + q 2n n=1 n=1 (5.0.14) These results thus form a set of four elements, all of which can be obtained by knowing just one of them; they are connected to one another as shown in the following scheme:

4qψ(−q 2 )ψ(−q 6 ) = θ5 (q)θ5 (q 3 ) = 4

∞ 

χ −4

[θ3 θ3 (q 3 ) − 1] ⇐ S ⇒ [θ4 θ4 (q 3 ) − 1] ⇐ P ⇒ θ2 θ2 (q 3 ) ⇐ S ⇒ θ5 θ5 (q 3 ), (5.0.15) where S and P are respectively the sign and Poisson operations. The Mellin transform also allows us to find relations amongst theta functions in a particularly simple fashion. Thus we have M[θ3 θ3 (q 3 ) − 1] = 2(1 + 21−2s )L 1 L −3 ,

(5.0.16)

M[θ2 θ2 (q 3 )] = 4(1 − 2−2s )L 1 L −3 , M[θ4 θ4 (q ) − 1] = −2(1 − 2 3

2−2s

(5.0.17)

)L 1 L −3 .

(5.0.18)

Hence M[θ3 θ3 (q 3 )] = M[θ2 θ2 (q 3 ) + θ4 θ4 (q 3 )]

(5.0.19)

θ3 θ3 (q 3 ) = θ2 θ2 (q 3 ) + θ4 θ4 (q 3 ),

(5.0.20)

and

which is Legendre’s classical result for the third-order modular relation. It is pertinent to find the Mellin transform of θ3 θ3 (q 3 ) − 1 directly from the definition of θ3 . Since θ3 θ3 (q 3 ) − 1 =

∞  ∞  −∞ −∞

qn

2 +3m 2

−1=



qn

2 +3m 2

,

(5.0.21)

Lattice sums and Ramanujan’s modular equations we have    M θ3 θ3 (q 3 ) − 1 = (n 2 + 3m 2 )−s = 2(1 + 21−2s )L 1 L −3 ,

191

(5.0.22)

 where  implies summation over all integer values of n and m but excluding the case when they are simultaneously zero. The reason for referring to such forms as θ3 θ3 (q 3 ) as two-dimensional may now be seen – namely, their Mellin transform yields a double sum over a two-dimensional lattice. In a similar fashion (5.0.9) and (5.0.10) may be treated in exactly the same way and from them two further modular forms of order 3 may be formed. Only four elements are generated from each of (5.0.7) and (5.0.9). The reason is that these elements exhibit a certain amount of symmetry, and so the S and P operations do not yield new results. Thus for example the Poisson transform of (5.0.8) yields itself. These eight results found from (5.0.7)–(5.0.10) are given in Table A.1 in the Appendix. Now, however, consider the following entries in C.19 (4.i)–(4.iv): qψ 5 (q)ψ(q 3 ) − 9q 2 ψ(q)ψ 5 (q 3 ) =

q 22 q 2 42 q 4 − + 1 − q2 1 − q4 1 − q8

52 q 5 − + ··· , (5.0.23) 1 − q 10  22 q 2 q 42 q 4 − 9φ(q)φ 5 (q 3 ) − φ 5 (q)φ(q 3 ) = 8 1 + + 2 1+q 1−q 1 − q4  52 q 5 − + ··· , (5.0.24) 1 + q5   q5 q ψ 3 (q) q7 q 11 − = 1 + 3 + − + · · · , 1−q 1 − q7 ψ(q 3 ) 1 − q5 1 − q 11   q2 φ 3 (q) q q4 q5 + =1+6 − − + ··· . 1−q φ(q 3 ) 1 + q2 1 + q4 1 − q5

(5.0.25)

(5.0.26)

Equations (5.0.23) and (5.0.24) are third-order modular equations of dimension 6 and (5.0.25) and (5.0.26) are third-order modular equations of dimension 2. Let us consider the latter. First, we put them into classical and character notation: ∞

 χ −6a (n)q n 1 θ2 3 (q 1/2 ) = 1 + 3 , 4 θ2 (q 3/2 ) 1 − qn n=1

∞

 χ −6b (n)q n θ3 3 = 1 + 6 . 1 + (−q)n θ3 (q 3 ) n=1

(5.0.27)

192

Lattice sums and Ramanujan’s modular equations

Now we apply alternate sign and Poisson transformations to obtain the following set of eight third-order modular equations of dimension 2: θ3 3 θ4 3 θ2 3 (q 3 ) θ5 3 (q 3 ) − 1 ⇐ S ⇒ − 1 ⇐ P ⇒ ⇐ S ⇒ θ2 θ5 θ3 (q 3 ) θ4 (q 3 ) ⇑ ⇑ P P ⇓ ⇓ θ5 3 θ4 3 (q 3 ) θ2 3 θ3 3 (q 3 ) ⇐S⇒ −1 ⇐S ⇒ −1 ⇐P ⇒ θ3 θ4 θ2 (q 3 ) θ5 (q 3 ) These results are displayed in Table A.2 together with their associated Mellin transforms. A similar analysis was carried out on (5.0.23) and (5.0.24) and again eight members of this set were found and are shown in Table A.3. Another approach to third-order modular equations was made by Borwein et al. [2]. Relations involving the expressons a(q), b(q), c(q), A = a(q 2 ), B = b(q 2 ), C = c(q 2 ), where  2 2 a(q) = q m +mn+n , (5.0.28)  2 2 b(q) = ωn−m q m +mn+n , (5.0.29)  2 2 c(q) = q (m+1/3) +(m+1/3)(n+1/3)+(n+1/3) , (5.0.30) with ω = exp(2πi/3), were considered. A connection between the work in [2] and that described here is briefly given. First, a(q) is given by its Lambert series as  ∞   q 3n+1 q 3n+2 , a(q) = 1 + 6 − 1 − q 3n+1 1 − q 3n+2 n=0

and it can be shown that θ4 3 (q) = 2a(q 2 ) − a(q) θ4 (q 3 )

and

θ4 3 (q 3 ) = 13 a(q) + 23 a(q 2 ). θ4 (q)

(5.0.31)

It is simple to show that these results agree with those given in Table A.2 by comparing their Mellin transforms. It is easily seen that M[a(q)−1] = 6L 1 (s)L −3 (s) and M[a(q 2 ) − 1] = 2−s 6L 1 (s)L −3 (s). So, from (5.0.31) we have

θ4 3 (q) − 1 = 21−s 6L 1 (s)L −3 (s) − 6L 1 (s)L −3 (s) M θ4 (q 3 ) M

= −6(1 − 21−s L 1 (s)L −3 (s),

θ4 3 (q 3 ) − 1 = 2L 1 (s)L −3 (s) + 4 × 2−s L 1 (s)L −3 (s) θ4 (q) = 2(1 + 21−s )L 1 (s)L −3 (s),

Lattice sums and Ramanujan’s modular equations

193

both of which agree with the results in Table A.2. For many other versions of third order modular equations the original paper [2] should be consulted. Two fifth-order equations of dimension 6 ((8.i) and (8.ii) in C.19) are given below. These are q 22 q 2 − 1 − q2 1 − q4 2 4 4 q 52 q 5 + − + ··· , 1 − q8 1 − q 10 q 2q 2 3q 3 qψ 3 (q)ψ(q 5 ) − 5q 2 ψ(q)ψ 3 (q 5 ) = − − 1 − q2 1 − q4 1 − q6 4q 4 6q 6 + + − ··· . 8 1−q 1 − q 12

qψ 5 (q)ψ(q 3 ) − 9q 2 ψ(q)ψ 5 (q 3 ) =

Treated as described above, a set of eight results was obtained and are given in Table A.4. In Table A.5 some seventh-order results, from Entry 17 in C.19, are given, and also relations for mixed modular equations of order 3, 5, and 15 derived from Entry 10 in C.20. It is evident from these examples how groups of four and eight members may be formed from any modular equation. It is manifest that if we are able to show that  (n 2 + 3m 2 )−s = 2(1 + 21−2s )L 1 L −3 , (5.0.32) in a way independent of Ramanujan’s methods then, by performing an inverse Mellin transform on (5.0.31), we immediately retrieve the modular equation θ3 θ3 (q ) − 1 = 2 3

∞ 

χ −3 q n

n=1

1 + (−q)n

.

Then, using the S and P operations, other modular relations are derivable. Now the systematic evaluation of the sums  S = S(a, b, c) = S(a, b, c : s) = (am 2 + bmn + cn 2 )−s ,  S1 = S1 (a, b, c) = S1 (a, b, c : s) = (−1)m (am 2 + bmn + cn 2 )−s ,  (−1)n (am 2 + bmn + cn 2 )−s , S2 = S2 (a, b, c) = S2 (a, b, c : s) =  S1,2 = S1,2 (a, b, c) = S1,2 (a, b, c : s) = (−1)m+n (am 2 + bmn + cn 2 )−s , where a, b, and c are integers, has been referred to in Section 1.4. It was conjectured there that if the binary quadratic form am 2 + bmn + cn 2 = (a, b, c) is such that its reduced forms are disjoint, i.e., there is just one reduced form per genus, then S is expressible as a linear sum of pairs of products of Dirichlet L-series. A prescription for finding these pairs of Dirichlet series was given by Zucker and Robertson [5]. Though counter-examples to this conjecture have been found,

194

Lattice sums and Ramanujan’s modular equations

they appear to be flukes. See Zucker and Robertson [6] for details. The principal solutions of all such cases of one reduced form per genus are given in Section 1.4. Thus the result  (n 2 + 7m 2 )−s = 2(1 − 21−s + 21−2s )L 1 L −7 is given there. By Mellin inversion this immediately gives θ3 θ3 (q ) − 1 = 2 7

∞  χ −14b q n n=1

1 + (−q)n

,

and then using S and P operations the other entries for seventh-order modular equations may be found, as shown in Table A.5. Again, using Mellin transforms it is simple to show that  θ3 θ3 (q 7 ) + θ4 θ4 (q 7 ) − θ2 θ2 (q 7 ) = 2θ4 (q 2 )θ4 (q 14 ) = 2 θ3 θ4 θ3 (q 7 )θ4 (q 7 ), or

 2 θ3 θ3 (q 7 ) + θ4 θ4 (q 7 ) = θ2 θ2 (q 7 ),

which is a well-known form of the seventh-order modular equation. Some quite high-order modular equations may also be deduced. Thus it was shown in [5] that if λ = (1+λ)/4 then for λ = 2, 3, 5, 11, 17, 41, corresponding to λ = 7, 11, 19, 43, 67, 163, S(1, 1, λ ) = 2L 1 L −λ . It was also shown that 2S(1, 1, λ ) = 22s [S(1, 0, λ) + S1,2 (1, 0, λ)]. Hence 22s [S(1, 0, λ) + S1,2 (1, 0, λ)] = 4L 1 L −λ . The inverse Mellin transform of the latter yields

θ3 (q 1/4 )θ3 (q λ/4 ) + θ4 (q 1/4 )θ4 (q λ/4 ) − 2 = 2 θ3 θ3 (q λ ) + θ2 θ2 (q λ ) − 1 =2

∞  χ −λ q n n=1

1 − qn

.

Mixed modular relations may also be similarly obtained. Thus from [5] we have  (m 2 + 15n 2 )−s = (1 − 21−s + 21−2s )L 1 L −15 + (1 + 21−s + 21−2s )L −3 L 5 , whilst from [6] we obtain  (3m 2 + 5n 2 )−s = (1 − 21−s + 21−2s )L 1 L −15 − (1 + 21−s + 21−2s )L −3 L 5 .

Lattice sums and Ramanujan’s modular equations

195

Adding these together yields   (m 2 +15n 2 )−s + (3m 2 +5n 2 )−s = 2(1−21−s +21−2s )L 1 L −15 . (5.0.33) Then, taking the inverse Mellin transform of (5.0.33) immediately gives the mixed modular equation of Entry (10.vi) in C.20. Once more the S and P operations yield the other results given in Table A.5. Entries (9.i–iv) in C.20 may then be simply deduced. Another example from [6] is S(1, 0, 14) + S(2, 0, 7) = L 1 L −56 + L −7 L 8 , which yields mixed modular equations of orders 2, 7, and 14 in the form θ3 θ3 (q 14 ) + θ3 (q 2 )θ3 (q 7 ) − 2 =

∞  χ −56 q n n=1

1 − qn

+

χ −7 (q n q 3n ) 1 + q 4n

.

Once again the use of S and P transformations would yield further results. It is clear from these examples that, given solutions to (5.0.32), many modular and mixed modular equations in terms of Lambert series may be generated. Since the Mellin transforms of modular equations give relations between multidimensional sums and Dirichlet L-series, such Mellin transforms may often be evaluated in terms of simple transcendental constants – a procedure of which Ramanujan, with his love of numbers, would surely have approved. For amusement we give some examples. A simple case is that of the fourth result of Table A.1, which may be translated as 4

∞ ∞   −∞ −∞

(−1)m+n  s = 4L −4 (s)L 12 (s). (2m − 12 )2 + 3(2n − 12 )2

√ √ For s = 1 the value of the right-hand side of the above is just π log(2 + 3)/ 3. Similarly, using the seventh result in Table A.1 we have, for the four-dimensional sum,  (−1)m+n+ p+r = −4(1 − 22−s )(1 − 31−s )L 1 (s)L 1 (s − 1). (m 2 + n 2 + 3 p 2 + 3r 2 )s (5.0.34) Putting s = 1 and s = 2 one has  (−1)m+n+ p+r = −2 log 3, m 2 + n 2 + 3 p 2 + 3r 2  (−1)m+n+ p+r 4 = − π 2 log 2. 2 2 2 2 2 9 (m + n + 3 p + 3r ) By differentiating both sides of (5.0.34) with respect to s, expressions such as    (−1)m+n+ p+r log(m 2 + n 2 + 3 p 2 + 3r 2 ) π = (2 log 3) log √ − γ , m 2 + n 2 + 3 p 2 + 3r 2 2 3 where γ is Euler’s constant, may be obtained.

196

Lattice sums and Ramanujan’s modular equations

A more complicated example is as follows. Let 

(−1)m+n+ p+r and + n 2 + p 2 + 5r 2 )s  (−1)m+n+ p+r B(s) = . 2 (m + 5n 2 + 5 p 2 + 5r 2 )s A(s) =

(m 2

Now, it is clear that   A(s) = M θ4 3 θ4 (q 5 ) − 1

and

  B(s) = M θ4 θ4 3 (q 5 ) − 1 ,

and, from Table A.4 we have A(s) − B(s) = −4(1 − 22−s )L 1 (s − 1)L 5 (s), 5B(s) − A(s) = −4(1 + 22−s )L 1 (s)L 5 (s − 1). Hence A(s) = −5(1 − 22−s )L 1 (s − 1)L 5 (s) − (1 + 22−s )L 1 (s)L 5 (s − 1), B(s) = −(1 − 22−s )L 1 (s − 1)L 5 (s) − (1 + 22−s )L 1 (s)L 5 (s − 1). For s = 1 and 2, A(s) and B(s) take particularly simple forms; we have the following results: 1 L 1 (0) = − , 2  √  1+ 5 2 L 5 (1) = √ log , 2 5

L 5 (0) = 0,



√  1 + 5 L 5 (0) = log , 2

π2 , 6 4π 2 L 5 (2) = √ , 25 5 L 1 (2) =

obtained using the standard limiting procedures √  1+ 5 . lim L 1 (s)L 5 (s − 1) = log s→1 2 

lim (1 − 2

1−s

s→1

)L 1 (s) = log 2,

Consequently, we have √  1+ 5 A(1) = −(3 + 5) log , 2 √     1 1+ 5 B(1) = − 3 + √ log , 2 5 √    1+ 5 −4π 2 log 2 1 + log , A(2) = √ 5 6 2 5 √    −4π 2 log 2 1 1+ 5 . B(2) = √ + log 25 6 2 5 √



Clearly many more such results may be established.

5.1 Commentary: The modular machine

197

5.1 Commentary: The modular machine As described in [2] there is a modular machine which, given sufficient knowledge of the ambient modular function theory, allows one to prove experimentally discovered equations of the kind described in this chapter. Various extensions are given in [3]. We paraphrase the modular approach as follows. It is possible to both find and prove modular identities entirely mechanically. We will illustrate how this works by proving the identity a 3 (q) = b3 (q) + c3 (q),

(5.1.1)

where, for ω = e2πi/3 , a(q) := b(q) := c(q) :=

∞ 

qn

n,m=−∞ ∞  n,m=−∞ ∞ 

2 +nm+m 2

ωn−m q n q (n+1/3)

,

2 +nm+m 2

,

2 +(n+1/3)(m+1/3)+(m+1/3)2

.

n,m=−∞

From the Lambert series  ∞   q 3n+1 q 3n+2 , a(q) = 1 + 6 − 1 − q 3n+1 1 − q 3n+2

(5.1.2)

n=0

we see that a(q) (in the variable τ , where q = e2πiτ ) is an Eisenstein series of weight 1 and character χ (d) = (d | 3) (the Legendre symbol modulo 3) for the congruence subgroup 0 (3). It is well known that if f (τ ) is a modular form on 0 (N ) then f (Mτ ) is a modular form on 0 (N M). From the equations 1 3 a(q 3 ) − a(q), 2 2 1 1 c(q) = a(q 1/3 ) − a(q), 2 2

b(q) =

(5.1.3) (5.1.4)

it follows that a(q 3 ), b(q 3 ), c(q 3 ), a(q 6 ), b(q 6 ), c(q 6 ) are all entire modular forms of weight 1 (and trivial character) on some congruence subgroup G, where (3) ∩ 0 (54) ⊂ G ⊂ 

198

Lattice sums and Ramanujan’s modular equations

(we are working in the variable q 3 to give c a Taylor series expansion at i∞). Here, as usual,    α β   := αδ − βγ = 1, α, β, γ , δ ∈ Z , γ δ       α β  (N ) := ∈   α ≡ δ ≡ 1 and β ≡ γ ≡ 0 mod N , γ δ      α β 0 (N ) := ∈   γ ≡ 0 mod N . γ δ The indices satisfy [ : (N )] = N 3

 p|N

and [ : 0 (N )] = N

1−

1  p2

 1 . 1+ p p|N

It follows that [ : G] ≤ [ : (3)] × [ : 0 (54)] ≤ 24 × 108. The standard theory can be found in [4]. Now suppose that P := P(a, b, c, A, B, C)

(5.1.5)

is a homogeneous polynomial of degree N in the six variables a := a(q), b := b(q), c := c(q), A := a(q 2 ), B := b(q 2 ), C := c(q 2 ). Then P(q) is an entire modular form of weight N on G and hence can have exactly N [ : G]/12 zeros in a fundamental region (counted in the local variables at the cusps). In particular P can have a zero of order at most 216N at τ = i∞. In other words, if the q-expansion of P vanishes through the first 216N + 1 terms then P ≡ 0. It is now a straightforward matter to generate a basis for all homogeneous identities of the type (5.1.5) for a fixed N . One expands the six functions a, b, c, A, B, C as q-series to some fixed order that is greater than the number of monomials in the expansion of (x1 + x2 + x3 + x4 + x5 + x6 ) N . One then solves the linear problem of finding a basis of identities to this fixed order. This must now be a superset of the desired identities. One then verifies that the q-expansion of each basis element vanishes through 216N + 1 terms, which proves that the alleged identity is a true identity (and not just an identity to a fixed number of terms). Since this is all done in exact integer arithmetic in a symbolic manipulation package this constitutes both a derivation and a proof. We will illustrate this for N = 3. What follows are the basis elements for all homogeneous cubic relations in a, b, c, A, B, C.

5.2 Commentary: A cubic theta function identity (1) A(c2 − aC − 2AC) (2) B(c2 − aC − 2AC) (3) c(bB − Aa + cC) (4) B(bB − Aa + cC) (5) C(bB − Aa + cC) (6) A(c A − ac + 2C 2 ) (7) B(c A − ac + 2C 2 ) (8) C(c A − ac + 2C 2 ) (9) −b(c A − ac + 2C 2 ) (10) B(b A − 2B 2 + ab) (11) C(b A − 2B 2 + ab) (12) −A(−a B − b2 + 2AB) (13) −B(−a B − b2 + 2AB) (14) −C(−a B − b2 + 2AB) (15) − B(Aa − 2bB − a 2 + 2A2 ) (16) − C(Aa − 2bB − a 2 + 2A2 )

(17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32)

199

−C 3 − B 3 + A3 4C 3 − 3acC + c3 2B 3 − 3b AB + b3 − bc A − bC 2 + cB 2 bcC − a B 2 + b A2 bc2 − b AC − 2B 2 C b2 c − acB + 4BC 2 ab2 − a AB − 2bB 2 + 2A2 B bcC + a 2 b − 3a B 2 + 2AB 2 2bBC − 3a AC + ac2 − 2A2 C 2bcB − 3ac A + a 2 c + 4AC 2 ab A − bcC + a B 2 − 2AB 2 bcB − ac A + aC 2 + 2AC 2 b AB − acC − 2B 3 + a 2 A −b AB − acC + 2C 3 + a A2 −3b AB − 3acC + 4C 3 + 2B 3 + a 3

Thus we find 32 basis elements, presented in factored form. Since there is a basis of six quadratic relations, many of them do factor. The verification of the identities requires computing Taylor series of length 650. The entire calculation took just a few minutes in Maple (in the early 1990s). Note that basis element (17) is our cubic modular equation (5.1.1) while basis element (3) is a quadratic identity given by Ramanujan. More careful analysis reveals the need to check only a lower-degree estimate in the q-expansion; however, since the computations are easy we have opted for the most straightforward estimates. Similar remarks apply for other forms and other N . Note that (1) and (12) combine to give 1 b2 1 c2 − , 2C 2 B while from (11) and (12) we may solve for A, B from a, b, and c. To summarize, the symbolic manipulation of series and products for theta functions – informed by the modular machine – is an enormously effective way of formalizing knowledge gleaned from numerical experimental tools such as PSLQ. a=

5.2 Commentary: A cubic theta function identity Let us return to a(q) =



qm

2 +nm+n 2

,

m,n

and c(q) =

b(q) =

 2 2 (ω)n−m q m +nm+n , m,n

 m,n

q (m+1/3)

2 +(n+1/3)(m+1/3)+(n+1/3)2

,

200

Lattice sums and Ramanujan’s modular equations

where ω = e2iπ/3 ; these expressed were given in (5.0.28). Then one may convince oneself computationally that for |q| < 1 the functions a, b, and c solve Fermat’s equation of degree 3, a 3 = b3 + c3 .

(5.2.1)

The neatest proof of (5.2.1) [2] relies on expressing b and c as q-products. For example, b(q) starts  3  2  3  3  2 1 − q3 1 − q4 1 − q5 1 − q6 (1 − q)3 1 − q 2 3  2 &3  % 1 − q8 1 − q9 , × 1 − q7 which tells us the closed form, whereas &21 % &345 % &8906 % &250257 % 1 − q4 1 − q6 1 − q8 1 − q2 a(q) = % &76 % &1734 % &46662 % &1365388 1 − q5 1 − q7 1 − q9 (1 − q)6 1 − q 3 clearly has no such nice product factorization. As sums, a and b look quite similar: a(q) = 1 + 6q + 6q 3 + 6q 4 + 12q 7 + 6q 9 + 6q 12 + 12q 13 + 6q 16   + 12q 19 + O q 21 and b(q) = 1 − 3q + 6q 3 − 3q 4 − 6q 7 + 6q 9 + 6q 12 − 6q 13 − 3q 16   − 6q 19 + O q 21 . It is instructive to write code to replicate these results and to explore c, which, like b, has both a nice q-product and a Lambert series. Note that a 3 = b3 + c3 is actually an identity for six-dimensional sums. Moreover, we have the following hypergeometric parameterization [2]:

1 2 , 3 b3 (q) 3 ; = a(q), 2 F1 1 a 3 (q) which implies a cubic transformation for the hypergeometric function that was known to Ramanujan.

References [1] B. C. Berndt. Ramanujan’s Notebooks Part III. Springer-Verlag, New York, 1991. [2] J. M. Borwein, P. B. Borwein, and F. G. Garvan. Some cubic modular identities of Ramanujan. Trans. Amer. Math. Soc., 343:35–47, 1994. [3] M. Hirschhorn, F. Garvan, and J. M. Borwein. Cubic analogues of the Jacobian theta function (q, z). J. Math. Phys., 19:1064, 1978.

References

201

[4] B. Schoeneberg. Elliptic Modular Functions. Springer-Verlag, New York, 1974. [5] I. J. Zucker and M. M. Robertson. Systematic approach to the evaluation of  2 2 −s (m,n=0,0) (am + bmn + cn ) . J. Phys. A, 9:1215–1225, 1976. [6] I. J. Zucker and M. M. Robertson. Further aspects of the evaluation of  2 2 −s (m,n=0,0) (am + bmn + cn ) . Math. Proc. Camb. Phil. Soc., 95:5–13, 1984.

6 Closed-form evaluations of threeand four-dimensional sums

In a 1980 survey of lattice sums [10], the following remarks may be found: ‘whereas in two dimensions it appears we know precisely when we can express a lattice sum exactly, and also in four and higher even dimensions where many exact results are obtainable, in contrast, in three dimensions there is a paucity of exact results. Indeed, only two are known . . . ’. The situation has changed somewhat since then and many more three-dimensional lattice sums have now been evaluated in closed form. These new results have been obtained by generalizing the Jacobian theta series and products described previously in Section 1.3. An account of these extensions is now given, and their properties described. Connections between series and products are discussed, and a notation developed to express them in a transparent way which enables them to be simply manipulated. Later in this chapter some recent developments on four-dimensional sums are also delineated.

6.1 Three-dimensional sums Four sets of series are considered (in this chapter, we often denote sums and prod  ucts simply by and respectively, when the summation or multiplication variable(s) are clear from the context). They are as follows: the set ,   2 2 θ¯ (k, l) = (−1)n q (kn+l) , (6.1.1) θ (k, l) = q (kn+l) ,   2 2 ¯ l) = φ(k, (−1)n(n+1)/2 q (kn+l) ; φ(k, l) = (−1)n(n−1)/2 q (kn+l) , the set X ,      2π n 2π n 2 (kn+l)2 n cos q q (kn+l) , χ (k, l) = , χ¯ (k, l) = (−1) cos 3 3    2π n 2 n(n−1)/2 cos q (kn+l) , ψ(k, l) = (−1) 3    2π n 2 n(n+1)/2 ¯ ψ(k, l) = (−1) cos q (kn+l) ; (6.1.2) 3

6.1 Three-dimensional sums the set  , θ  (k, l) =

203

  2 2 (kn + l)q (kn+l) , θ¯  (k, l) = (−1)n (kn + l)q (kn+l) ,  2 φ  (k, l) = (−1)n(n−1)/2 (kn + l)q (kn+l) , (6.1.3)  2 φ¯  (k, l) = (−1)n(n+1)/2 (kn + l)q (kn+l) ;

the set X  ,      2π n 2π n 2  (kn+l)2  n (kn+l)q q (kn+l) , χ (k, l) = cos , χ¯ (k, l) = (−1) cos 3 3    2π n 2 (kn + l)q (kn+l) , (−1)n(n−1)/2 cos (6.1.4) ψ  (k, l) = 3    2π n 2 ψ¯  (k, l) = (−1)n(n+1)/2 cos (kn + l)q (kn+l) . 3 In all these series denotes summation over all integers n from −∞ to ∞. All the previous Jacobian theta series are accommodated by this notation; thus θ3 = θ (1, 0) = θ (1, 1),

θ4 = θ¯ (1, 0),

θ2 = θ (2, 1)|q 1/4 ,

θ5 = 2θ¯ (4, 1)|q 1/4 . Here the vertical bar followed by q t means that q t replaces q in the original expression. These series have expansion properties, typically θ (k, l) = θ (2k, l) + θ (2k, k + l) = θ (3k, l) + θ (3k, k + l) + θ (3k, 2k + l) + · · · θ¯ (k, l) = θ (2k, l) − θ (2k + k, l) = θ¯ (3k, l) − θ¯ (3k, k + l) + θ¯ (k, 2k + l) + · · · ¯ φ(k, l) = θ¯ (2k, l) + θ(2k, k + l), ¯ ¯ φ(k, l) = θ (2k, l) − θ(2k, k + l).

(6.1.5)

These expansions will allow us sometimes to convert new series into more conventional notation. For example, expanding θ (1, 1) we have θ (1, 1) = θ (3, 1) + θ (3, 2) + θ (3, 3). But θ (3, 1) = θ (3, 2) and θ (3, 3) = θ (1, 1)|q 3 = θ3 (q 9 ), and we thus have  1 θ3 − θ3 (q 9 ) . θ (3, 1) = 2 We further define two infinite products   ¯ Q(k, l) = (1 − q kn−l ), Q(k, l) = (1 + q kn−l ), (6.1.6)  where implies the product over all n from 1 to ∞. These also have expansion properties; thus Q(k, l) = Q(2k, l)Q(2k, 2k, k + l) = Q(3k, l)Q(3k, k + l)Q(3k, 2k + l) = · · · (6.1.7)

204

Closed-form evaluations of three- and four-dimensional sums

¯ with a similar expression for Q(k, l). It may be shown that all members of  ¯ may be expressed in terms of Q(k, l) and Q(k, l). For example, starting with the fundamental Jacobi identity   2 (1 − q 2n )(1 + xq 2n−1 )(1 + x −1 q 2n−1 ), xnqn = 2

2

replacing q by q k , x by q 2kl , and multiplying both sides by q l , one obtains immediately ¯ ¯ k + 2l) Q(2k, k − 2l)|q k . θ (k, l) = q l Q(2k, 0) Q(2k, 2

(6.1.8)

With similar but more complicated substitutions we may deduce that 2 θ¯ (k, l) = q l Q(2k, 0)Q(2k, k + 2l)Q(2k, k − 2l)|q k ,

¯ ¯ φ(k, l) = q l Q(4k, 0) Q(4k, 2k)Q(4k, k − 2l) Q(4k, 3k − 2l) 2

¯ Q(4k, k + 2l)Q(4k, 3k + 2l)|q k , ¯ ¯ ¯ l) = q l 2 Q(4k, 0) Q(4k, 2k) Q(4k, k − 2l)Q(4k, 3k − 2l)Q(4k, k + 2l) φ(k, ¯ Q(4k, 3k + 2l)|q k . Some further notational additions are introduced here. The Euler partition func tion Q(k, 0) = (1−q kn ) will be denoted by (k) and Jacobi’s favourite products, Q0 = Q3 =

 

(1 − q 2n ),

Q1 =



(1 + q 2n ),

Q2 =



(1 + q 2n−1 ), (6.1.9)

(1 − q 2n−1 ),

will be denoted by w, x, z, y respectively. A dictionary is easily established between the two sets of notations; thus w = (2),

x=

(4) , (2)

y=

(1) , (2)

z=

(2)2 , (1)(4)

(6.1.10)

from which the well-known result x yz = 1 is immediately obtained. Expressions for  and X in terms of (k) will be termed Eulerian, whereas expressions in terms of w, x, y, z will be called Jacobian. It is also convenient to introduce here a slight modification of the Euler partition function, namely the Dedekind eta function, which is defined as  η(k) = q k/24 (1 − q kn ) = q k/24 (k) := [k]. Three operations which will be applied to sums and products introduced here will now be described. The first is the operation S (the sign transform), which changes q to −q in any expression. We have the following results:

6.1 Three-dimensional sums Sw = w,

S x = x,

S y = z,

S(k) = (k) for k an even integer;

S(k) =

205

Sz = y; (2k)3

(k)(4k)

(6.1.11)

for k an odd integer.

(See above for the meaning of (k).) For example, these results lead to the following: Sθ3 = θ4 ,

S

θ2 (q 1/2 ) θ5 (q 1/2 ) = . q 1/8 q 1/8

The second operation, P, is a Poisson transformation (cf. the Poisson summation formula). This transformation changes expressions in term of q = e−π t into those in terms of a complementary variable, ρ = e−π/t . The fundamental result for a q-series is (   2π nl 1 2 2 cos ρ n /k . Pkθ (k, l) = (6.1.12) t k The other thetas transform as below: (   πl 1 2 2 ¯ Pk θ (k, l) = cos (2n + 1) ρ (2n+1) /4k , (6.1.13) t k (    π k + 2l 1  2 2 cos(2n + 1) cos(2n + 1)π ρ (2n+1) /16k , Pkφ(k, l) = t 4 4k (6.1.14) (      π k + 2l 1 2 2 ¯ l) = Pk φ(k, sin (2n + 1) sin(2n + 1)π ρ (2n+1) /16k . t 4 4k (6.1.15) For transforming products the most important result is expressed in terms of the Dedekind eta function. This transforms as follows: (   2 1 Pη[2k : q] = η :ρ , (6.1.16) kt k which enables one to transform any result in Eulerian form to another in that form, since the P operation is multiplicatively transitive. That is, P[a][b][c] · · · = P[a]P[b]P[c] · · · . The Jacobian quantities in (6.1.10) transform thus:     P q 1/12 x = 2−1/2 ρ −1/24 y, P q 1/12 w = ρ 1/12 w,     (6.1.17) P q −1/24 z = ρ 1/24 z. P q −1/24 w = 21/2 ρ 1/12 x, Expressions related by P-transformations will be termed Poisson conjugates. Since q and ρ are completely interchangeable, a Poisson transform of a q-relation will yield another q-relation unless the expression is self-conjugate.

206

Closed-form evaluations of three- and four-dimensional sums

The third operation which will be applied is the Mellin transform, M, defined by  ∞ 1 t s−1 f (t) dt. M[ f (t)] = (s) 0 This operation can be applied to any q-series after replacing q by e−t . As examples of such a transformation, consider first  2 2 2 2 θ¯ (6, 1) = (−1)n q (6n+1) = q − q 5 − q 7 + q 11 + · · · Replacing q by e−t and taking the Mellin transform, we have  ∞ 1  2 ¯ 1) = (−1)n M θ(6, t s−1 e−t (6n+1) dt (s) 0  (−1)n = = 1 − 5−2s − 7−2s + 11−2s + · · · = L 12 (2s), (6n + 1)2 where L 12 is a positive-parity Dirichlet L-series of argument 2s. Similarly, if we perform the same operations on θ¯  (6, l), we have  2 2 2 2 θ¯  (6, 1) = (−1)n (6n + 1)q (6n+1) = q + 5q 5 − 7q 7 − 11q 11 + · · · , M θ¯  (6, 1) = 1 + 52s−1 − 72s−1 − 112s−1 + · · · = (1 + 31−2s )L −4 (2s − 1), where L −4 is a negative-parity L-series of argument 2s − 1. In general, Mellin transforms of members of  and X always yield positive-parity L-series of degree 2s, whereas Mellin transforms of members of  and X  give negative-parity L-series of degree 2s − 1. The reason why this preliminary discussion was presented was to explain how members of , X ,  , and X  may be expressed in a particularly apt form of infinite product. This will be illustrated with the example of θ (3, 1). From (6.1.8), ¯ ¯ θ (3, 1) = q Q(6, 0) Q(6, 5) Q(6, 1)|q 3 . Now ¯ ¯ ¯ ¯ ¯ ¯ ¯ Q(1, 0) = Q(6, 0) Q(6, 1) Q(6, 2) Q(6, 3) Q(6, 4) Q(6, 5), so ¯ ¯ Q(6, 5) Q(6, 1) =

¯ Q(1, 0) . ¯ ¯ ¯ ¯ Q(6, 0) Q(6, 2) Q(6, 3) Q(6, 4)

But (4) ¯ ¯ ¯ ¯ ¯ ¯ ¯ 0)|q 2 = Q(6, 0) Q(6, 2) Q(6, 4) = Q(3, 0) Q(3, 1) Q(3, 2)|q 2 = Q(1, (2) and (2) ¯ , Q(1, 0) = (1)

¯ ¯ Q(6, 3) = Q(2, 1)|q 3 =

(6)2 . (3)(12)

6.1 Three-dimensional sums

207

Combining these together leads to θ (3, 1) = q

(6)2 (9)(36) (3)(12)(18)

or

θ (3, 1)|q 1/3 = q 1/3

(2)2 (3)(12) . (6.1.18) (1)(4)(6)

Now perform a sign transform on both side of (6.1.18); thus S

θ¯ (3, 1)|q 1/3 (2)2 (3)(12) θ (3, 1)|q 1/3 = = S (1)(4)(6) q 1/3 q 1/3

and, following the prescription given in (6.1), one obtains θ¯ (3, 1)|q 1/3 = q 1/3

(1)(6)2 . (2)(3)

Similarly, apply the Poisson transform to θ (3, 1). From (6.1.12), 1  1 2π n n 2 /9 Pθ (3, 1) = √ = √ χ (1, 0)|ρ 1/9 . cos ρ 3 3 t 3 t From (6.1.18) we have

  '  '   2     1

2

1

2

4

1

3

9

9 18 3 9 9 [6]2 [9][36] 1 3 Pθ (3, 1) = P = √ '  '  '        . [3][12][18] t 2 1 1 4 1 2 3

6

9

3

Equating these two expressions for Pθ (3, 1) and replacing ρ 1/9 by q, we obtain χ (1, 0) =

(1)(4)(6)2 . (2)(3)(12)

¯ 1) and a This process may be continued by taking a Poisson transform of θ(3, sign transform of χ (1, 0) and then alternating these two transforms, forming the following closed set: θ(3, 1)|q 1/3 ⇑

⇐S⇒

θ¯ (3, 1)|q 1/3

⇐P⇒

2χ(2, 1)|q 1/8

⇐S⇒

P

⇓ χ(1, 0)

¯ 2ψ(2, 1)|q 1/8 ⇑ P

⇐S⇒

χ(1, ¯ 0)

⇐P⇒

θ(6, 1)|q 1/24

⇐S⇒

⇓ φ(6, 1)|q 1/24

Following the procedures just described, all the members of sets  and X given in Table 6.1 in Eulerian and Jacobian forms were constructed. In Table 6.2 the sign, Poisson, and Mellin transforms of these series are also given. As previously described, the Mellin transforms of all the series in Table 6.1 yield a Dirichlet L-series whose character depends on the series. Because L-series of different character are algebraically independent, the independence or otherwise of the various elements of  and X may easily be established. Since the history of expressing infinite series in product form and vice versa goes back to Euler, it has not been possible to determine with certainty the original

208

Closed-form evaluations of three- and four-dimensional sums Table 6.1 Members of  and of X which may be expressed in Eulerian and Jacobian forms

Series

Conventional

Eulerian

Jacobian

Equation

θ (1, 0)

θ3

wz 2

T(1.1)

¯ 0) θ(1,

θ4

wy 2

T(1.2a)

¯ 0)|q 2 θ(1,

θ4 (q 2 )

wyz

T(1.2b)

θ (2, 1)|q 1/8 = 2θ (4, 1)|q 1/8

θ2 (q 1/2 )

(2)5 (1)2 (4)2 (1)2 (2) (2)2 (4) 2 2q 1/8 (2) (1)

2q 1/8 wx z

T(1.3a)

θ (2, 1)|q 1/4 = 2θ (4, 1)|q 1/4 ¯θ(4, 1)|q 1/8

θ2

2q 1/4 (4) (2)

2q 1/4 wx 2

T(1.3b)

1 1/2 ) 2 θ5 (q   1 1/3 ) − θ (q 3 ) 3 2 θ3 (q   1 3 1/3 ) 2 θ4 (q ) − θ4 (q   1 6 2/3 ) 2 θ4 (q ) − θ4 (q   1 1/6 ) − θ (q 3/2 ) 2 2 θ2 (q

q 1/8 (1)(4) (2)

q 1/8 wx y

T(1.4)

q 1/3 z|wx y|q 3

T(1.5)

q 1/3 y|wx z|q 3

T(1.6a)

q 2/3 yz|wx 2 |q 3

T(1.6b)

q 1/24 x z|wy 2 |q 3

T(1.7a)

q 1/12 x|wyz|q 3

T(1.7b)

θ (3, 1)|q 1/3

2

q 1/3

(2)2 (3)(12) (1)(4)(6)

2 q 1/3 (1)(6) (2)(3) 2 q 1/3 (2)(12) (4)(6) 2 q 1/24 (2)(3) (1)(6) 2 q 1/12 (4)(6) (2)(12) 5 q 1/24 (1)(4)(6) (2)2 (3)2 (12)2 q 1/24 (1)

q 1/24 x y|wz 2 |q 3

T(1.8)

q 1/24 wy

T(1.9a)

¯ 1)|q 1/12 θ(6,

q 1/12 (2)

q 1/12 w

T(1.9b)

¯ 1)|q 1/6 θ(6,

q 1/6 (4)

q 1/6 wx

T(1.9c)

¯ 1)|q 1/24 φ(6,

(2)3 q 1/24 (1)(4) (1)(4)(6)2 (2)(3)(12) (2)2 (3) (1)(6) (4)2 (6) (2)(12) 2 (6) q 1/8 (1) (2)(3) 2 (12) q 1/4 (2) (4)(6) 2 q 1/8 (2)2 (3)(12) (1) (4)2 (6)2

q 1/24 wz

T(1.10)

wx y|z|q 3

T(1.11)

wx z|y|q 3

T(1.12a)

wx 2 |yz|q 3

T(1.12b)

q 1/8 wy 2 |x z|q 3

T(1.13a)

q 1/4 wyz|x|q 3

T(1.13b)

q 1/8 wz 2 |x y|q 3

T(1.14)

¯ 1)|q 1/3 θ(3, ¯ 1)|q 2/3 θ(3, θ (6, 1)|q 1/24 θ (6, 1)|q 1/12 φ(6, 1)|q 1/24 ¯ 1)|q 1/24 θ(6,

χ (1, 0) χ(1, ¯ 0) χ(1, ¯ 0)|q 2 2χ (2, 1)|q 1/8 2χ (2, 1)|q 1/4 ¯ 2ψ(2, 1)|q 1/8

  1 1/3 ) − θ (q 3 ) 2 2 θ2 (q   1 θ (q 1/3 ) + θ (q 3 ) 5 2 5





1 3θ (q 9 ) − θ 3 3 2   1 3θ (q 9 ) − θ 4 4 2   1 18 2 2 3θ4 (q ) − θ4 (q )

  1 1/2 ) − 3θ (q 9/2 ) 2 2 θ2 (q   1 9 2 θ2 − 3θ2 (q )   1 1/2 ) + 3θ (q 9/2 ) 5 2 θ5 (q

sources of the results given in Table 6.1. However, after a considerable literature search it is believed that the following attributions of the results in Table 6.1 are correct. Result (T1.l) is due to Jacobi; (T1.2) and (T1.3) are due to Gauss; (T1.9) is Euler’s classic result, the very first of its kind; (T1.6) and (T1.7) were given by Kac [14], who claims them as new and also gives (T1.11) and (T1.13) as new, but those results go back to Ramanujan [18]. Results (T1.4) and (T1.10) as given here are new, though (T1.4) is implied by [24]; however, [22] quotes some results of Jacobi from which (T1.4) and (T1.l0) may be deduced. As far as we know

6.1 Three-dimensional sums

209

Table 6.2 The sign, Poisson, and Mellin transforms of some members of  and X ; the argument of the L-series is (2s) Series

Sign transform Poisson transform

Mellin transform

Equation

θ (1, 0) − 1

T(2.1)

¯ 0) − 1 θ(1,

θ (1, 0) − 1

2L 1

¯ 0) − 1 θ(1,

θ (1, 0) − 1

θ (2, 1)|q 1/4

θ (2, 1)|q 1/8

¯ 1)|q 1/8 2θ(4,

√ ¯ 0)|q 2 2θ(1,

−2(1 − 21−2s )L

θ (3, 1)|q 1/3

¯ 1)|q 1/3 θ(3,

¯ 1)|q 1/3 θ(3,

T(2.2)

1

21+3s (1 − 2−2s )L 1

T(2.3)

3−1/2 χ (1, 0)|q 1/3

3s (1 − 3−2s )L

T(2.4)

θ (3, 1)|q 1/3

2 × 3−1/2 χ (2, 1)|q 1/12

3s (1 − 21−2s )(1 − 3−2s )L 1

T(2.5)

¯ 1)|q 1/8 θ(4,

θ (4, 1)|q 1/8

8s L

T(2.6)

θ (6, 1)|q 1/2

φ(6, 1)|q 1/24

¯ 1)|q 1/24 θ(6,

¯ 1)|q 1/24 φ(6,

¯ 1)|q 1/8 θ(4, √ −1 6.3 χ¯ (1, 0)|q 2/3 √ ¯ 1)|q 1/6 2θ(6,

φ(6, 1)|q 1/24

θ (6, 1)|q 1/24

¯ 3−1/2 ψ(2, 1)|q 1/24

24s (1 + 3−2s )L

¯ 1)|q 1/24 φ(6,

¯ 1)|q 1/24 θ(6,

¯ 1)|q 1/24 φ(6,

24s L 24

χ (1, 0)

χ(1, ¯ 0)

θ (3, 1)|q 1/9

−(1 − 31−2s )L

χ(1, ¯ 0)

χ (1, 0)

θ (6, 1)|q 1/36

(1 − 21−2s )(1 − 31−2s )L 1

2χ (2, 1)|q 1/8

¯ 2ψ(2, 1)|q 1/8

¯ 1)|q 2/9 θ(3,

8s (1 − 2−2s )(1 − 31−2s )L

¯ 2ψ(2, 1)|q 1/8

2χ (2, 1)|q 1/8

φ(6, 1)|q 1/72

8s (1 + 31−2s )L 8

1

8

24s (1 − 2−2s )(1 − 3−2s )L 1

T(2.7)

24s L 12

T(2.8) T(2.9)

8

T(2.10) T(2.11)

1

T(2.12) 1

T(2.13) T(2.14)

(T1.5), (T1.8), (T1.12), and (T1.14) are new as given here, though [2] establishes something equivalent to (T1.12) in terms of conventional notation. In Table 6.3 we display results for  and X  in Eulerian and Jacobian form. It has not been found possible to obtain these results by the systematic procedure used for  and X . Ad hoc methods were needed to derive the results given in Table 6.3. One result, (T3.3), may be derived from a classical identity of Jacobi, which is given in [12] as ∞   (−1)n (2n + 1)q n(n+1)/2 = (1 − q n )3 = (1)3 .

(6.1.19)

n=0

A more symmetric form of (6.1.19) may be found by using the fact that, for any function f, ∞ 

(−1)n (2n + 1) f (2n + 1) =

 (4n + 1) f (4n + 1).

n=0

Thus, multiplying both sides of (6.1.19) by q 1/8 the latter may be written  2 (4n + 1)q (4n+1) /8 = q 1/8 (1)3 . A method of finding  in terms of Lambert series and recognizing these as expressible as infinite products can produce desired results. This will be illustrated ¯  (k, 1). The Jacobi triple product identity may be written now for 

210

Closed-form evaluations of three- and four-dimensional sums   2 (−1)n a n q n = (2) (1 − aq 2n−1 )(1 − a −1 q 2n−1 ).

(6.1.20)

The following operations are carried out in this order: (i) replace a by a k ; (ii) multiply both sides of (6.1.20) by a l ; (iii) differentiate logarithmically with respect to 2 a, then multiply by a; (iv) replace q by q k and a by q 2l . Then it is found that   2 2 2 2  θ¯  (k, l) q 2nk −k −2kl q 2nk −k +2kl =1+k − . (6.1.21) 2 2 2 2 θ¯ (k, l) 1 − q 2nk −k −2kl 1 − q 2nk −k +2kl Series such as those found on the right-hand side of (6.1.21) are known as Lambert series. In order to identify them, Mellin transforms are formed. For example, consider k = 4, and 1 = 1. We have ∞   q 4n−3 q 4n−1 θ¯  (4, 1) 1/8 − . (6.1.22) |q − 1 = 4 1 − q 4n−3 1 − q 4n−1 θ¯ (4, 1) 1

The Mellin transform of the right-hand side of (6.1.22) is 4L 1 (s)L −4 (s). But it is a well-known result [12], going back to [16] and implicit in Jacobi, that M[θ32 − 1] = M[θ 2 (1, 0) − 1] = 4L 1 (s)L −4 (s). So, the right-hand side of (6.1.22) is just θ 2 (1, 0) − 1, and thus ¯ 1)|q 1/8 θ 2 (1, 0) = q 1/8 θ¯  (4, 1)|q 1/8 = θ(4,

(2)9 , (1)3 (4)3

which after using the results (T1.1) and (T1.4) gives (T3.4). Table 6.3 Members of  and X  expressed in Eulerian and Jacobian forms; the argument of the L-series is (2s − 1) Series

Eulerian

Jacobian

Mellin transform

Equation

θ  (3, 1)|q 1/3

q 1/3 w 3 x 2 y 2

3s L −3

T(3.1)

q 1/3 w 3 x 2 z 2

3s (1 + 22−2s )L −3

T(3.2)

θ  (4, 1)|q 1/8

2 (4)2 q 1/3 (1)(2) 5 q 1/3 (2)2 (1) q 1/8 (1)3

q 1/8 w 3 y 3

23s L −4

T(3.3)

θ¯  (4, 1)|q 1/8

q 1/8

(2)9 (1)3 (4)3 5 q 1/24 (1)2 (2) 13 q 1/24 (2) (1)5 (4)5 q 1/8 (1)3 + 3q 9/8 (9)Â cˇ

q 1/8 w 3 z 3

23s L

T(3.4)

q 1/24 w 3 y 5

24s (1 + 21−2s )L −3

T(3.5)

q 1/24 w 3 z 5

24s L −24

T(3.6)

θ¯  (3, 1)|q 1/3

θ  (6, 1)|q 1/24 φ  (6, 1)|q 1/24 θ¯  (6, 1)|q 1/8 2χ¯  (2, 1)|q 1/8

9 9 − 3q 9/8 (18) q 1/8 (2) (1)3 (4)3 (9)3 (36)3 q 1/8 (1)3 + 9q 9/8 (9)3

2ψ  (2, 1)|q 1/8

q 1/8

φ¯  (6, 1)|q 1/8

9 (2)9 − 9q 9/8 (18) (1)3 (4)3 (9)3 (36)3

−8

23s (1 + 31−2s )L

−4

T(3.7)

23s (1 − 31−2s )L −8

T(3.8)

23s (1 + 32−2s )L

−4

T(3,9)

23s (1 − 32−2s )L

−8

T(3.10)

6.1 Three-dimensional sums As a further example, consider k = 3 and l = 1. From (6.1.21), ∞   q 6n−5 q 6n−1 θ¯  (3, 1) 1/3 |q =1+3 − . 1 − q 6n−5 1 − q 6n−1 θ¯ (3, 1)

211

(6.1.23)

1

Now, the right-hand side of (6.1.23) is Entry (4.iii) in Chapter 19 of Ramanujan’s Notebooks Part III [1] and is given in his notation. Translating this into classical notation, it is just θ23 (q 1/2 )/4θ2 (q 3/2 ). Using (T1.3a) this becomes (3)(2)6 /(1)3 (6)2 ; then, multiplying by (T1.6a) we obtain (T3.2). A sign transform of (T3.2) yields (T3.1) and a Poisson transform of (T3.2) gives (T3.5). Finally a sign transform of the latter gives (T3.6). These six results appear to be the only ones in which members of  can be expressed as a single product of Eulerian terms. In attempting to find others using (6.1.21), invariably a sum of two terms appears. Thus, for k = 6, l = 1, ∞   q 3n−2 q 3n−1 θ¯  (6, 1) 1/24 −1=6 − . |q 1 − q 3n−2 1 − q 3n−1 θ¯ (6, 1) 1

The Mellin transform of the right-hand side of the latter is 6L 1 (s)L −3 (s), which is also the Mellin transform of θ3 θ3 (q 3 ) + θ2 θ2 (q 3 ) − 1 [27]. Hence   θ¯  (6, 1)|q 1/24 = θ¯ (6, 1)|q 1/24 θ3 θ3 (q 3 ) + θ2 θ2 (q 3 ) . (6.1.24) This is then the sum of two Eulerian products, and it does not seem possible to collapse these into a single form. Furthermore, many different combinations of the two terms can be made because many relations exist amongst theta functions. Thus the modular equation of order 3 may be expressed in terms of theta functions as θ3 θ3 (q 3 ) = θ2 θ2 (q 3 ) + θ4 θ4 (q 3 ). So, (6.1.24) could be written as   θ¯  (6, 1)|q 1/24 = θ¯ (6, 1)|q 1/24 2θ2 θ2 (q 3 ) + θ4 θ4 (q 3 ) , resulting in a completely different combination of Eulerian terms. The simplest form is obtained by inverting the Mellin transforms obtained from the expansions of  and X  , and this is how (T3.7)–(T3.10) were obtained. It is noticeable that the Mellin transforms of θ  (3, l), θ¯  (3, 1), and θ  (6, 1), which all contain L −3 , should be related but this is not evident from their Eulerian forms. However, it may be simply demonstrated from two classical results for theta functions. The first is θ3 = θ3 (q 4 ) + θ2 (q 4 ), which, if written in Eulerian form, is (2)5 (8)5 (16)2 . = + 2q 2 2 2 2 (8) (1) (4) (4) (16)

212

Closed-form evaluations of three- and four-dimensional sums

Multiplying both sides by q 1/3 (4)2 gives q 1/3

5 2 (2)5 1/3 (8) 4/3 (4)(16) , = q + 2q (8) (1)2 (16)2

i.e., θ¯  (3, 1)|q 1/3 = θ  (6, 1)|q 1/3 + 2θ  (3, 1)|q 4/3 . Similarly, from the classical result 2θ3 (q 4 ) = θ3 + θ4 we obtain 2θ¯  (6, 1)|q 1/3 = θ  (3, 1)|q 1/3 + θ¯  (3, 1)|q 1/3 . The results in Table 6.3 also have a long history. As previously mentioned (T3.3) goes back to Jacobi. Results (T3.1) and (T3.5) are usually ascribed to [11]. However, they go back at least to Ramanujan [18], who quotes them along with Jacobi’s result without reference, as though they were well known. However, formulae given by Ramanujan without comment imply several possibilities. It may be that (a) he did not know a reference, (b) he thought the formulae too well known to need a reference, (c) the formulae were his own, or (d) any combination of the above. Thus (T3.1) and (T3.5) may well have originated with Ramanujan, and up to now no earlier source has been found. Result (T3.2) is given by [14] and is simply a sign transform of (T3.1). Result (T3.4) or its equivalent seems to have appeared first in [10] and is simply a sign transform of (T3.3). Result (T3.6) as presented here is new. It was suggested in [2] that performing the equivalent of a sign transformation on what was essentially  (6, 1) would yield an interesting relation. The result is in fact φ  (6, 1) and (T3.6). Results (T3.7)–(T3.10) are new. (These remarks were made by Zucker [26]. Almost simultaneously a paper was published by Köhler [15] in which (T3.1), (T3.2), and (T3.4)–(T3.6) were given and derived in a completely different manner. Since Köhler’s paper was received by his journal on 16 January 1989 and Zucker’s on 22 March 1989, Köhler should be credited with (T3.6). Both authors were in complete ignorance of one another.) It will be observed from Tables 6.1 and 6.3 that the members of  and of X contain w to a single power whereas the members of  and of X  contain w 3 . It is thus possible to combine three suitable members of  and X to form a members of  or of X  . For example, remembering that x yz = 1, θ (1, 0)θ¯ (1, 0)θ (4, 1)|q 1/4 = q 1/4 wz 2 wy 2 wx 2 = q 1/4 w 3 = θ  (4, 1)|q 1/4 . In classical notation this is Jacobi’s expression θ2 θ3 θ4 = θ1 . Written out fully this is   2 2 2 2 (4n + 1)q (4n+1) /4 . (−1)m q m +n +(4 p+1) /4 = Now replacing q by e−t and taking Mellin transforms of both sides, we obtain  −s  = 4s L −4 (2s − 1). (6.1.25) (−1)m m 2 + n 2 + (4 p + 1)2 /4

6.1 Three-dimensional sums

213

Thus the three-dimensional sum on the left-hand side of (6.1.25) has been expressed in closed form, as was pointed out in [8]. Another example is

3 θ¯ (6, 1) = θ  (4, 1)|q 3 , or 

 −s (−1)m+n+ p (6m + 1)2 + (6n + 1)2 + (6 p + 1)2 = 3−s L −4 (2s − 1).

(6.1.26)

This result was given by [7]. It should be noticed that (6.1.26) is apparently the first three-dimensional Coulomb sum to have been evaluated in closed form and that for, s = 1/2, (6.1.26) is the first such analytic result of direct physical significance (it is the electrostatic potential at six points in the unit cell for NaCl). It is clear, then, that any combination of results in Tables 6.1 and 6.2 Table 6.4 Three-dimensional sums evaluated in terms of a single Dirichlet  L-series. Here implies summation over all indices m, n, p from −∞ to ∞; A = L −4 (2s − 1), B = L −3 (2s − 1), C = (1 + 22−2s )L −3 (2s − 1), D = (1 + 21−2s )L −3 (2s − 1), E = L −8 (2s − 1)  (−1)m [4m 2 + 4n 2 + (4 p + 1)2 ]−s = A  (−1)m [4m 2 + 4n 2 + (2 p + 1)2 ]−s = 2A  (−1)m+n [8m 2 + 8n 2 + (4 p + 1)2 ]−s = A  (−1)m+n [8m 2 + 8n 2 + (2 p + 1)2 ]−s = 2A  (−1)m+n+ p [8m 2 + 16n 2 + (4 p + 1)2 ]−s = A  (−1)m [8m 2 + (4n + 1)2 + (4 p + 1)2 ]−s = 2−s A  (−1)m [8m 2 + (4n + 1)2 + (2 p + 1)2 ]−s = 21−s A  (−1)m+n [(4m + 1)2 + (4n + 1)2 + 8 p 2 ]−s = 2−s A  (−1)m+n [16m 2 + (4n + 1)2 + (4 p + 1)2 ]−s = 2−s A  (−1)m+n [16m 2 + (4n + 1)2 + (2 p + 1)2 ]−s = 21−s A  (−1)m+n+ p [(6m + 1)2 + (6n + 1)2 + (6 p + 1)2 ]−s = 3−s A  (−1)m+n+ p [24m 2 + (6n + 1)2 + 2(6 p + 1)2 ]−s = 3−s A  (−1)m [(4m + 1)2 + (4n + 1)2 + 2(4 p + 1)2 ]−s = 2−2s A  (−1)m [(4m + 1)2 + (2n + 1)2 + 2(4 p + 1)2 ]−s = 21−2s A  (−1)m [(4m + 1)2 + (4n + 1)2 + 2(2 p + 1)2 ]−s = 21−2s A  (−1)m [(4m + 1)2 + (2n + 1)2 + 2(2 p + 1)2 ]−s = 22−2s A  (−1)m+n [(6m + 1)2 + 2(6n + 1)2 + 3(4 p + 1)2 ]−s = 6−s A  (−1)m+n [(6m + 1)2 + 2(6n + 1)2 + 3(2 p + 1)2 ]−s = 2 × 6−s A  (−1)m+n+ p( p+1)/2 [3(4m + 1)2 + 2(6n + 1)2 + (6 p + 1)2 ]−s = 6−s A  (−1)m+n+ p( p+1)/2 [(6m + 1)2 + 4(6n + 1)2 + (6 p + 1)2 ]−s = 6−s A  (−1)m [4(3m + 1)2 + 4(3n + 1)2 + (6 p + 1)2 ]−s = 9−s A

cont.

214

Closed-form evaluations of three- and four-dimensional sums Table 6.4 (cont.)

 (−1)m+n [72m 2 + (6n + 1)2 + 8(3 p + 1)2 ]−s = 9−s A  (−1)m [8(3m + 1)2 + 9(4n + 1)2 + (6 p + 1)2 ]−s = 18−s A  (−1)m+n(n−1)/2 [16(3m + 1)2 + (6n + 1)2 + (6 p + 1)2 ]−s = 18−s A  (−1)m+n(n−1)/2 [9(4m + 1)2 + (6n + 1)2 + 8(3 p + 1)2 ]−s = 18−s A  [2 − (−1)m+n ](−1) p [24m 2 + 72n 2 + (6 p + 1)2 ]−s = (1 + 31−2s )A  [1 + (−1)m+n ](−1) p [2m 2 + 6n 2 + (6 p + 1)2 ]−s = 2(1 + 32−2s )A  [2 − (−1)m+n ](−1) p [8m 2 + 24n 2 + (6 p + 1)2 ]−s = (1 + 32−2s )A  [1 + (−1)m+n ](−1) p [6m 2 + 18n 2 + (6 p + 1)2 ]−s = 2(1 + 31−2s )A  (−1)m cos 2π3m cos 2π3 n cos 2π3 p [4m 2 + 4n 2 + (2 p + 1)2 ]−s = 2−1 A  (−1)m+n cos 2π3 n cos 2π3 p [8m 2 + 8n 2 + (2 p + 1)2 ]−s = 2−1 A  (−1)m cos 2π3m cos 2π3 n [8m 2 + (2n + 1)2 + (4 p + 1)2 ]−s = 2−1−s A  (−1)m cos 2π3m cos 2π3 n [8m 2 + (2n + 1)2 + (2 p + 1)2 ]−s = 2−s A  (−1)m+n(n+1)/2 cos 2π3 n cos 2π3 p [(4m + 1)2 + (2n + 1)2 + 8 p 2 ]−s = 2−1−s A  (−1)m+n(n+1)/2 cos 2π3m cos 2π3 n cos 2π3 p [16m 2 + (2n + 1)2 + (2 p + 1)2 ]−s = 2−2−s A  (−1)m+n+ p [3m 2 + 12n 2 + (6 p + 1)2 ]−s = B  (−1)m+n+ p [6m 2 + (6n + 1)2 + (6 p + 1)2 ]−s = 2−s B  (−1)m+n [12m 2 + (6n + 1)2 + 3(4 p + 1)2 ]−s = 4−s B  (−1)m+n [12m 2 + (6n + 1)2 + 3(2 p + 1)2 ]−s = 2 × 4−s B  (−1)m+n+ p( p+1)/2 [12m 2 + 3(4n + 1)2 + (6 p + 1)2 ]−s = 4−s B  (−1)m+n [(6m + 1)2 + (6n + 1)2 + 6(4 p + 1)2 ]−s = 8−s B  (−1)m+n [(6m + 1)2 + (6n + 1)2 + 6(2 p + 1)2 ]−s = 2 × 8−s B  (−1)m+n+ p [3(4m + 1)2 + (4n + 1)2 + 2(6 p + 1)2 ]−s = 8−s B  (−1)m [(6m + 1)2 + 3(4n + 1)2 + 12(4 p + 1)2 ]−s = 16−s B  (−1)m [(6m + 1)2 + 3(4n + 1)2 + 12(2 p + 1)2 ]−s = 2 × 16−s B  (−1)m+n(n+1)/2+ p [3(4m + 1)2 + (6n + 1)2 + 12(4 p + 1)2 ]−s = 16−s B  (−1)m [(6m + 1)2 + 3(4n + 1)2 + 12(2 p + 1)2 ]−s = 2 × 16−s B  (−1)m+n(n+1)/2+ p [3(4m + 1)2 + (6n + 1)2 + 12(2 p + 1)2 ]−s = 2 × 16−s B  (−1)m [2(6m + 1)2 + 3(8n + 1)2 + 3(8 p + 1)2 ]−s = 32−s B  (−1)m+n+ p( p+1)/2 [3(8m + 1)2 + 3(8n + 3)2 + 2(6 p + 1)2 ]−s = 32−s B  (−1)m+n [12m 2 + (6n + 1)2 + 3 p 2 ]−s = C  (−1)m+n [(6m + 1)2 + (6n + 1)2 + 6 p 2 ]−s = 2−s C  (−1)m [(6m + 1)2 + 12n 2 + 3(4 p + 1)2 ]−s = 4−s C  (−1)m [(6m + 1)2 + 12n 2 + 3(2 p + 1)2 ]−s = 2 × 4−s C  (−1)m [2(6m + 1)2 + 3(4n + 1)2 + 3(4 p + 1)2 ]−s = 8−s C  (−1)m [2(6m + 1)2 + 3(4n + 1)2 + 3(2 p + 1)2 ]−s = 2 × 8−s C  (−1)m+n(n+1)/2 [4(6m + 1)2 + (6n + 1)2 + 3(4 p + 1)2 ]−s = 8−s C  (−1)m(m+1)/2+n(n+1)/2 [(6m + 1)2 + (6n + 1)2 + 6(4 p + 1)2 ]−s = 8−s C

cont.

6.1 Three-dimensional sums

215

Table 6.4 (cont.)  (−1)m(m+1)/2+n(n+1)/2 [(6m + 1)2 + (6n + 1)2 + 6(2 p + 1)2 ]−s = 2 × 8−s C  (−1)m+n(n+1)/2 [4(6m + 1)2 + (6n + 1)2 + 3(2 p + 1)2 ]−s = 2 × 8−s C  (−1)m+n+ p [24m 2 + 24n 2 + (6 p + 1)2 ]−s = D  (−1)m+n [12m 2 + (6n + 1)2 + 12 p 2 ]−s = D  (−1)m+n [(6m + 1)2 + (6n + 1)2 + 24 p 2 ]−s = 2−s D  (−1)m+n+ p( p+1)/2 [48m 2 + (6n + 1)2 + (6 p + 1)2 ]−s = 2−s D  (−1)m+n(n+1)/2+ p( p+1)/2 [24m 2 + (6n + 1)2 + (6 p + 1)2 ]−s = 2−s D  (−1)m [(6m + 1)2 + 48n 2 + 3(4 p + 1)2 ]−s = 4−s D  (−1)m [(6m + 1)2 + 48n 2 + 3(2 p + 1)2 ]−s = 2 × 4−s D  (−1)m+n(n+1)/2 [3(4m + 1)2 + (6n + 1)2 + 48 p 2 ]−s = 4−s D  (−1)m(m−1)/2+n(n+1)/2+ p( p+1)/2 [3(4m + 1)2 + 3(4n + 1)2 + 2(6 p + 1)2 ]−s = 8−s D  (−1)m [(4m + 1)2 + 8n 2 + 8 p 2 ]−s = E  (−1)m [16m 2 + 8n 2 + (4 p + 1)2 ]−s = E  (−1)m [16m 2 + 8n 2 + (2 p + 1)2 ]−s = 2E  (−1)m+n(n+1)/2 [2(6m + 1)2 + (6n + 1)2 + 24 p 2 ]−s = 3−s E  (−1)m(m+1)/2+n(n+1)/2+ p( p+1)/2 [(6m + 1)2 + (6n + 1)2 + (6 p + 1)2 ]−s = 3−s E  (−1)m(m−1)/2 [(6m + 1)2 + 72n 2 + 8(3 p + 1)2 ]−s = 9−s E      (−1)m(m−1)/2 cos 2π3m cos 2π3 n [(2m + 1)2 + 8n 2 + 8 p 2 ]−s = 2−1 E  2[(−1)m+n − 1](−1) p ( p + 1)/2[24m 2 + 72n 2 + (6 p + 1)2 ]−s ] = (1 − 31−2s )E  2[(−1)m+n − 1](−1) p ( p + 1)/2[8m 2 + 24n 2 + (6 p + 1)2 ]−s ] = (1 − 32−2s )E  (−1)m(m+1)/2 [(6m + 1)2 + 24n 2 + 24 p 2 ]−s = L −24 (2s − 1)

which yield a result in Table 6.4 will lead to the evaluation of a three-dimensional sum in terms of a single Dirichlet L-series. In Table 6.4 a list of nearly 80 such evaluations is given. It is not known whether this list is exhaustive or whether any given result is a trivial consequence of another. Several results in Table 6.3 that are not immediately obvious from the entries in Tables 6.1 and 6.2 have been obtained as sign or Poisson transforms of other results. For integer s all these three-dimensional sums can be expressed √ exactly in terms of powers of π and surds, since L −d (2s − 1) = Rπ 2s−1 d, where R is a rational number; see [27]. Although an attempt has been made to obtain these results in a consistent fashion, it is evident that complete success has not been achieved. The evaluation of three-dimensional sums seems to depend on finding  as a single term in Eulerian form, and no systematic way of doing this is known or is necessarily available. The numerical factors 24 also appear to play a central role in these matters. For example, it is not possible to put θ (5, 1) into Eulerian form. Clearly the processes

216

Closed-form evaluations of three- and four-dimensional sums

described might be carried further. For example one could define  2 θ¯  (k, l) = (kn + l)2 q (kn+l) . There would be no difficulty in finding the Mellin transforms of such quantities. Thus  2 2 2 2 2 (6n + 1)2 q (kn+1) = q 1 − 52 q 5 − 72 q 7 + 112 q 11 + · · · θ¯  (6, 1) = and M θ¯  (6, 1) = 1 − 52s−2 − 72s−2 + 112s−2 + · · · = L 12 (2s − 2), and the Mellin transforms of members of  and X  obviously yield positiveparity L-series of order 2s − 2. However, the problem of obtaining  as a single Eulerian product seems even more intractable than that for  , and so far no success with any  has been achieved. If Eulerian or Jacobian expressions for  could be obtained then certain five-dimensional sums could be found in closed form, and the progression to higher odd dimensions is obvious. However, only one five-dimensional sum, with a quadratic form but lacking the squared terms, is known in closed form [9] (see Commentary 6.3 after the next section). This was derived from a q-series based on manipulation of a basic hypergeometric series. This method has not yet been explored further.

6.2 Four-dimensional sums Shortly after the publication of the work in Chapter 1, the short paper [7] appeared; this used the theta function method to obtain a variety of intriguing sums. One is based on a q-series of Jacobi [13], which can be formulated as ∞

 √ (−1)n (2n + 1)q n(n+1)/2 . Q 30 ( q) =

(6.2.1)

n=0

By setting q = e−2t/3 , multiplying$ (6.2.1) on both sides by any inverse Laplace ∞ transform F[s], such that f (s) = 0 e−st F(t) dt, and integrating over s from 0 to ∞, one obtains the reduction formula   ∞     (n + 12 )2 . (−1)n 1 +n 2 +n 3 f || Rn + ( 16 , 16 , 16 )||2 = (−1)n (2n + 1) f 3 Rn

n=0

(6.2.2) (Here Rn = (n 1 , n 2 , n 3 ) and the sum is over all integer values −∞ < n i < ∞.) 1/2 In particular, for the choice f (s) = s −1/2 e−as , the sum on the right-hand side of (6.2.2) is elementary and yields the Yukawa potential sum     √ e−a|| Rn +(1/6,1/6,1/6)|| a = . (6.2.3) (−1)n 1 +n 2 +n 3 3 sech √ n + ( 1 , 1 , 1 )|| || R 12 6 6 6  Rn

6.2 Four-dimensional sums

217

√ The value in the Coulomb limit a → 0+ is 3. This appears to be the known value that is ‘closest’ to the Madelung constant and, other than at points where the potential vanishes by symmetry, it remains the only Coulomb sum known exactly for an actual crystal. A second result worth quoting is the n-fold sum ∞ 

···

l1 =1

∞ 

(−1)l1 +···+ln

ln =1

ρn (−1)n = n , sinh πρn 2 (2n + 1)π

(6.2.4)

' where ρn = l12 + · · · + ln2 . This is the n-dimensional analogue of the sum (1.5.7), which was evaluated by contour integration. Next we examine a fourdimensional analogue of (6.2.3) and see that it has revealing connections with the fascinating subject of elliptic curves and modular functions. The four-dimensional analogue of the shifted rocksalt series (6.1.26) is S(s) =

∞ 

(−1)n 1 +n 2 +n 3 +n 4

n j =−∞

[(n 1 + 16 )2 + (n 2 + 16 )2 + (n 3 + 16 )2 + (n 4 + 16 )2 ]s

.

(6.2.5)

Its evaluation takes us into new aspects of theta functions and q-series and even displays a connection with Wiles’ famous resolution of Fermat’s last theorem. We begin in the usual way by exponentiating the denominator to get ∞

4  ∞  1 2 dt t s−1 (−1)n e−t (n+1/6) . (6.2.6) S(s) = (s) 0 n=−∞ By Euler’s pentagonal number identity ∞ 

(−1)n q k(6n+1)

2 /24

= q k/24

n=−∞

∞ 

(1 − q kn ),

(6.2.7)

n=1

where q = e−2t/3 , one has S(s) =

( 23 )s (s)



∞

dt t s−1 η4

η(τ ) = e2iπ τ/24

∞ 

1 3

 it

(6.2.8)

(1 − e2nπiτ )

(6.2.9)

0

where



n=1

is the Dedekind eta function. In passing we note that [17] η4 (6τ ) is the modular function associated with the elliptic curves of conductor 36, as required by the restricted Taniyama–Shimura conjecture proved by A. Wiles. This will be commented on below. By means of the Euler and Jacobi q-series given in [17] for η and its cube, we find  ∞ ∞ ∞  ( 2 )s  2 2 (−1)m+n (2m + 1) dt t s−1 e−[(6n+1) +3(2m+1) ]/36 . S(s) = 3 (s) n=−∞ 0 m=0 (6.2.10)

218

Closed-form evaluations of three- and four-dimensional sums

Next, by evaluating the integral in (6.2.10) we obtain the double-series reduction of (6.2.5): S(s) = 12s

∞  ∞ 

(−1)m+n

n=−∞ m=0

[(6n

2m + 1 . + 3(2m + 1)2 ]s

+ 1)2

(6.2.11)

When s is an integer the m-series in (6.2.11) can be evaluated in terms of hyperbolic functions, yielding a single-series reduction of S(s). Thus, we find ∞ ∞ 4π  4π  (−1)n χ (n) π S(1) = √ (6.2.12) (−1)n sech √ (6n +1) = √ √ 6 n=−∞ 6 n=1 cosh(nπ/ 12) 12

where χ (2n) = χ (3n) = 0 and χ (6n ± 1) = 1. Since ∞ 

(−1)(n+1)/2 √ cosh(π n/ 12) n odd equals K /k for a certain modulus, S(1) is elliptic in nature and, one expects, has a hypergeometric representation. This has been proved explicitly for S(2) by Rogers and Zudilin [21] and by Rodriguez-Villegas [19], who found that

  4 5 , , 1, 1 2π 2 ; 12 . (6.2.13) ln 54 − 19 4 F3 3 3 S(2) = 81 2, 2, 2 At this point it is appropriate to summarize briefly several key mathematical notions. An elliptic curve E, over the rational numbers, say, can be described by an equation of the form y 2 = x 3 + ax + b with a, b ∈ Z and has a well-defined tangent at each point (x, y). Then the discriminant D = 16(4a 3 + 27b2 ) = 0. If E has rational points R, R  , the line through R and R  must intersect E at some point (including the point O at infinity) (r1 , r2 ). Addition is defined by R + R  = (r1 , −r2 ) with respect to which the set of rational points forms an abelian group, which, by the Mordell–Weil theorem, has the form Z r ⊕T . Furthermore, Mazur’s theorem states that the torsion group T must be either Z/mZ (m = 1, 2, . . . , 10 or 12) or Z/2Z ⊕ Z/2mZ (m = 1, 2, 3, 4). Determining the rank r is complicated. If the prime p does not divide D then, with respect to the finite field F p = {0, 1, . . . , p − 1}, E will consist of ν p points and we define a p = 1 + p − ν p . √ Hasse’s theorem states that |a p | ≤ 2 p. The Hasse–Weil L-function is defined by −1  ∞  ap an 1 1 − s + 2s−1 = , L(E, s) = p ns p p

(6.2.14)

n=1

where the second equality in (6.2.14) defines an for non-prime n. Three major results are:

6.2 Four-dimensional sums Weil–Wiles theorem expression

219

There is an integer N (the conductor) such that the  (E, s) =

N 4π 2

s/2 (s)L(E, s)

continues analytically to an entire function of s satisfying the functional equation (E, s) = ±(E, 1 − s). Modularity conjecture

(6.2.15)

The function f (z) =

∞ 

an einz

(6.2.16)

n=1

is modular. Birch–Swinnerton-Dyer conjecture

We have

L(E, s) = O((s − 1)r )

as s → 1.

(6.2.17)

Wiles famously proved the modularity conjecture for certain conductors; values and references can be found in [17]. A one-million-dollar prize still awaits a proof of the third item. The conductor N is closely related to the product of the primes dividing the discriminant, so is an invariant. From our discussion of S(s) we have, for conductor N = 36, that the modular function f is η4 and therefore L(E 36 , s) =

1 S(s). 9s

(6.2.18)

The Mahler measure of a multivariate polynomial P(x1 , x2 , . . . , xk ) is defined by  d t ln |P(e2πit1 , . . . , e2πitk )|. (6.2.19) m(P) = [0,1]k

The evaluation of the Mahler measures arising in our context in hypergeometric form is quite non-trivial and has been carried out by Rodriguez-Villegas [19], and Rogers and Zudilin [21]. In 1997 Deninger [6] conjectured that m(1 + x + 1/x + y + 1/y) = (15/4π 2 )L(E 15 , 2). Subsequently Boyd [3], guided by numerical results, conjectured over 100 such relations for various conductors, many lying outside the class considered by Wiles and all for s = 2. In their seminal work [21], Rogers and Zudilin put these in terms of the class of lattice sums F(a, b, c, d) = ∞ 

(−1)n 1 +n 2 +n 3 +n 4 (a + b + c + d)2

. 2 2 2 2 2 n j =−∞ a(6n 1 + 1) + b(6n 2 + 1) + c(6n 3 + 1) + d(6n 4 + 1) (6.2.20)

220

Closed-form evaluations of three- and four-dimensional sums

By means of these connections, to date the following hypergeometric evaluations have been proved:

  4 5 , 3 , 1, 1 1 2π 2 1 3 ; 2 F(1, 1, 1, 1) = , (6.2.21) ln 54 − 9 4 F3 81 2, 2, 2

1 1 1 π2 2, 2, 2 1 ; 2 , F(1, 1, 2, 2) = √ 3 F2 (6.2.22) 1, 32 8 2

  4 5 , 3 , 1, 1 4π 2 1 1 3 ; −8 F(1, 1, 3, 3) = . (6.2.23) ln 6 + 108 4 F3 81 2, 2, 2 Some examples of the conjectured and since proved (see e.g. [21]) evaluations are:

  4 5 , , 1, 1 3π 2 5 1 3 3 ; 27 ln 2 − 16 F(1, 1, 5, 5) = , (6.2.24) 4 F3 40 3 2, 2, 2 32

1 1 1 π2 2, 2, 2 1 ; 4 , F(1, 2, 3, 6) = (6.2.25) 3 F2 12 1, 32

1 1 1 π2 2, 2, 2 1 ; 16 . F(1, 3, 5, 15) = (6.2.26) 3 F2 15 1, 32 For further examples, details, and references see [20, 21]. It is clear that much of this material is worthy of further investigation.

6.3 Commentary: A five-dimensional sum Whereas many closed forms for lattice sums in two, four, six, and eight dimensions may be found – see [2, 23, 25] – results for odd dimensions are notoriously difficult to establish. Indeed, in Chapter 1 it was noted that only two exact results for three-dimensional lattice sums were known when the material in that chapter was first published. That so many more three-dimensional sums have now been found in closed form is possibly an aberration: none of them can be linked to any physical lattice. It is noteworthy that the closed-form results for three-dimensional lattice sums appear as a single L-series. For even dimensions one always find pairs of products of two L-series. In five dimensions it seems only one result has been established [9]. This is ∞ 

(m 1 m 2 + m 2 m 3 + m 3 m 4 + m 4 m 1 + m 2 m 5 )−s

m 1 ,m 4 =0 m 2 ,m 3 ,m 5 =1

= ζ (s)ζ (s − 2) − ζ 2 (s − 1), and again it is expressed as sums of pairs of L-series.

(6.3.1)

6.4 Commentary: A functional equation for a three-dimensional sum 221 However, it is possible to find lattice sums which result in the product of four L-series. Thus, if  2  m 2 + n 2 + p 2 + q 2 + mp + nq ζ K (s) = (m,n, p,q=0,0,0,0)

−3 (mq + np + nq)2

s

and √ √ n + 12 p 3 + 12 q ≥ 0, mq + np + nq ≥ 0, m − p − n 3 > 0,   m 2 + n 2 + p 2 + q 2 + mp + nq − 3 (mq + np + nq) > 0 then ζ K (s) = L 1 (s)L −3 (s)L −4 (s)L 12 (s). This was found by Shail and Zucker (1985, unpublished). The summand in ζ K is the norm of the numbers represented by the imaginary bicyclic √ biquadratic√field √ formed by the adjunction of the four fields N, Q( −3), Q( −4), and Q( 12). This particular field has class number 1. Details of such fields may be found in [5]. Brown and Parry [4] showed that there are just 47 imaginary bicyclic biquadratic fields with class number 1; hence more results are available similar to the one given above. Thanks are due to M. Peters (Wilhelms-Universität Münster) for providing this information.

6.4 Commentary: A functional equation for a three-dimensional sum In Section 1.2, and again in Commentary 3.9, the Mellin transform, the Poisson summation formula, and Hobson’s integral were used together to write some lattice sums as rapidly convergent series. Here, we use this technique to re-derive the functional equation (1.3.42) for a three-dimensional lattice sum  % &−s m 2 + n 2 + p2 . a(2s) = m,n, p

First, the Mellin transform gives a(2s) =

 1   ∞ s−1 −t (m 2 +n 2 + p2 ) t e dt. (s) m,n, p 0

We can then proceed in two ways: apply the Poisson summation formula to a single sum, say, over m, or apply it twice to a pair of sums – say over n and p. Thus, we produce

222

Closed-form evaluations of three- and four-dimensional sums a(2s) = =

√  1 π + 2s (s) m m  n, p





m; n, p=0 0

1 π + (s) (n 2 + p 2 )s

∞



t s−3/2 e−t (n 

∞

2 + p 2 )−π 2 m 2 /t

t s−2 e−tm

dt

2 −π 2 (n 2 + p 2 )/t

(6.4.1) dt.

m=0; n, p 0

(6.4.2) Using Hobson’s integral  ∞  q s/2 √ t s−1 e− pt−q/t dt = 2 K s (2 pq), p 0 which is valid for Re p, Re q > 0, we can develop the above expressions into triple sums over the modified Bessel function of the second kind; terms which cannot be evaluated using Hobson’s integral are treated separately, resulting in line and plane sums. Evaluating the plane sums using the result  % &−s m 2 + n2 = 4ζ (s)L −4 (s), C(s) := m,n

equations (6.4.1) and (6.4.2) reduce to the expressions √ π (s − 12 ) a(2s) = 2ζ (2s) + C(s − 12 ) (s)  s−1/2 ∞ ' % & m 4π s    K s−1/2 2π m n 2 + p 2 , (6.4.3) + (s) n 2 + p2 m=1 n, p a(2s) =

2π ζ (2s − 2) + C(s) s−1  2  ∞ ' % & 4π s   n + p 2 s−1 + K s−1 2π m n 2 + p 2 . (s) m n, p

(6.4.4)

m=1

Consider now the transformation s → 32 − s applied to (6.4.3): the summand in (6.4.3) transforms into the summand in (6.4.4), since K ν (z) = K −ν (z). Substituting the transformed equation into (6.4.4), we obtain a relation between a(2s) and a(3 − 2s). This relation can be simplified using the reflection formulae of the ζ, L −4 , and  functions and the duplication formula of the latter; in the end we obtain a(3 − 2s)( 32 − s) a(2s)(s) , (6.4.5) = πs π 3/2−s which is equation (1.3.42) and provides an analytic continuation for a(2s). Note that this process can be generalized to higher dimensions, and it provides a rich source of sum equalities for K s ; the equality of (6.4.3) and (6.4.4) is but one example. Similar equations are derivable for alternating sums.

6.5 Commentary: Two amusing lattice sum identities

223

6.5 Commentary: Two amusing lattice sum identities (1) Show that, for k = 0, ∞ 

(−1)

n

n=0

sinh πk (2n + 1)

cosh2 πk (2n + 1) − cos2

πl k

∞ (−1)n k πl  = sec . 4 k n=−∞ cosh π2 (kn + l)

Proof We start with the double sum ∞ ∞  

(−1)m+n

n=−∞ m=0

2m + 1 . (kn + l)2 + (2m + 1)2

The n-summation is tabulated in [7]: sinh πk (2m + 1) cos πl π k k(2m + 1) cosh2 πk (2m + 1) − cos2

πl k

.

Similarly, the m-summation is π π sech (kn + l). 4 2 Hence, equating the two remaining sums we have the stated result. (2) Another amusing evaluation, which can be traced back to Liouville, is ∞ 

(n 2 n 1 ,n 2 ,n 3 ,n 4 =0 1

+ n 22

+ n 23

+ n 24

1 + n 1 + n 2 + n 3 + n 4 + 1)s

= (1 − 2−s )(1 − 21−s )ζ (s)ζ (s − 1), valid for Re s > 2. Proof The sum can be written as ∞  n 1 ,n 2 ,n 3 ,n 4  ∞

= Ms

1 % &s 1 2 1 2 (n 1 + 2 ) + (n 2 + 2 ) + (n 3 + 12 )2 + (n 4 + 12 )2 =0 q

(n 1 +1/2)2

n 1 =0

=

 4 1 2

∞  n 2 =0

q

(n 2 +1/2)2

∞  n 3 =0

q

(n 3 +1/2)2

∞ 

q

(n 4 +1/2)2



n 4 =0

Ms [θ24 ],

where as usual Ms denotes the Mellin transform. Now the result follows; see [23].

224

Closed-form evaluations of three- and four-dimensional sums

References [1] B. C. Berndt. Ramanujan’s Notebooks Part III. Springer-Verlag, New York, 1991. [2] J. M. Borwein and P. B. Borwein. Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity. Wiley, New York, 1987. [3] D. W. Boyd. Mahler’s measure and special values of L-functions. Exp. Math., 7:37– 82, 1998. [4] E. Brown and C. J. Parry. The imaginary bicyclic biquadratic fields with class number 1. J. Reine Angew. Math., 266:118–120, 1974. [5] H. Cohn. A Classical Invitation to Algebraic Numbers and Class Fields. SpringerVerlag, Berlin and New York, 1978. [6] C. Deninger. Deligne periods of mixed motives, K -theory and the entropy of certain Z n -actions. J. Amer. Math. Soc., 10:259–281, 1997. [7] P. J. Forrester and M. L. Glasser. Some new lattice sums including an exact result for the electrostatic potential within the NaCl lattice. J. Phys. A, 15:911–914, 1982. [8] M. L. Glasser. The evaluation of lattice sums. I. Analytic procedures. J. Math. Phys., 14:409–413, 1973. [9] M. L. Glasser. The evaluation of lattice sums. IV. A five-dimensional sum. J. Math. Phys., 16:1237–1238, 1975. [10] M. L. Glasser and I. J. Zucker. Lattice sums. In Theoretical Chemistry, Advances and Perspectives (H. Eyring and D. Henderson, eds.), vol. 5, pp. 67–139, 1980. [11] B. Gordon. Some identities in combinatorial analysis. Quart. J. Math., 12:285–290, 1961. [12] G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers, 4th edition. Clarendon, Oxford, 1960. [13] C. G. Jacobi. Fundamenta Nova Theoriae Functionum Ellipticarum. Konigsberg, 1829. [14] V. G. Kac. Infinite-dimensional algebras, Dedekind’s η-function, classical Möbius function and the very strange formula. Adv. Math., 30:85–136, 1978. [15] G. Köhler. Some eta-identities arising from theta series. Math. Scand., 66:147–154, 1990. [16] L. Lorenz. Bidrag tiltalienes theori. Tidsskrift Math., 1:97–114, 1871. [17] Y. Martin and K. Ono. Eta-quotients and elliptic curves. Proc. Amer. Math. Soc., 125:3169–3176, 1997. [18] S. Ramanujan. On certain arithmetical functions. Trans. Camb. Phil. Soc., 22:159– 184, 1916. [19] F. Rodriguez-Villegas. Modular Mahler measures I. In Topics in Number Theory, pp. 17–48. Kluwer, Dordrecht, 1999. [20] M. Rogers. Hypergeometric formulas for lattice sums and Mahler measures. IMRN, 17:4027–4058, 2011. [21] M. Rogers and W. Zudilin. From L-series of elliptic curves to Mahler measures. Compositio Math., 148:385–414, 2012. [22] H. J. S. Smith. Report on the Theory of Numbers. Chelsea, New York, 1865. [23] I. J. Zucker. Exact results for some lattice sums in 2, 4, 6 and 8 dimensions. J. Phys. A, 7:1568–1575, 1974.

References

225

[24] I. J. Zucker. New Jacobian θ functions and the evaluation of lattice sums. J. Math. Phys., 16:2189–2191, 1975. [25] I. J. Zucker. Some infinite series of exponential and hyperbolic functions. SIAM J. Math. Anal., 15:406–413, 1984. [26] I. J. Zucker. Further relations amongst infinite series and products II. The evaluation of 3-dimensional lattice sums. J. Phys. A, 23:117–132, 1990. [27] I. J. Zucker and M. M. Robertson. Systematic approach to the evaluation of  2 2 −s (m,n=0,0) (am + bmn + cn ) . J. Phys. A, 9:1215–1225, 1976.

7 Electron sums

In 1934 Wigner [17] introduced the concept of an electron gas bathed in a compensating background of positive charge as a model for a metal. He suggested that under certain circumstances the electrons would arrange themselves in a lattice, and that the body-centred lattice would be the most stable of the three common cubic structures. Fuchs [14] appears to have confirmed this in a calculation on copper relying on physical properties of copper. The evaluation of the energy of the three cubic electron lattices under precise conditions was carried out by Coldwell-Horsfall and Maradudin [10] and became the standard form for calculating the energy of static electron lattices, U (lattice). In this jellium model, electrons are assumed to be negative point charges located on their lattice sites and surrounded by an equal amount of positive charge uniformly distributed over a cube centred at the lattice point. First, one calculates the interaction energy, U1 , of a single electron with all the other electrons on their lattice sites. Then one finds the energy of interaction of an electron with the compensating positive background, U2 . Thus U (lattice) = U1 − U2 .

(7.0.1)

The procedure is outlined here with the simple cubic lattice as its paradigm. Essentially one attempts to evaluate (7.0.1), where U1 =

1 e 2  , a0 (m 2 + n 2 + p 2 )1/2

U2 =

e2 a0



∞ ∞ ∞ 0

0

0

d x d y dz . (x 2 + y 2 + z 2 )1/2 (7.0.2)

In (7.0.2) e is the charge on an electron, a0 is the side of the elementary cube of the lattice, and  is the sum over all integers m, n, p from −∞ to ∞ excluding the case when m, n, p are simultaneously zero. Now both U1 and U2 are divergent quantities, so the approach is to find ways of evaluating them so that the infinities

Electron sums

227

cancel. An outline of this procedure is given now; the common factor e2 /a0 is ignored. The energy U1 is first expressed as an integral; thus     ∞ U1 = t −1/2 exp −π(m 2 + n 2 + p 2 )t dt. 0

Next, we break the range of integration into two parts, (0,1) and (1, ∞), and define     ∞ U11 = t −1/2 exp −π(m 2 + n 2 + p 2 )t dt, U12 =

 

1 1

  t −1/2 exp −π(m 2 + n 2 + p 2 )t dt.

0

In terms of the auxiliary integrals φk , which are given by  ∞ φk (x) = t k e−xt dt,

(7.0.3)

1

U11 can be written as follows:    φ−1/2 π(m 2 + n 2 + p 2 ) . U11 =

(7.0.4)

For U12 , remove the restriction on the sum by subtracting the m = n = p = 0 term, thus obtaining  1  1   −1/2 2 2 2 t exp −π(m + n + p )t dt − t −1/2 dt. (7.0.5) U12 = 0

0

In (7.0.5) use the Poisson transform ∞ 

exp(−π m 2 t) =

−∞

  ∞  1 −π m 2 √ exp t t −∞

and perform the second integral, to obtain  1  π  t −2 exp − (m 2 + n 2 + p 2 ) dt − 2. U12 = t 0

(7.0.6)

In (7.0.6) restore the restriction on the sum by adding back the m = n = p = 0 term and, in the remaining integral, substitute t = 1/u to obtain  1     ∞ dt exp −π u(m 2 + n 2 + p 2 ) du − 2 + . (7.0.7) U12 = 2 1 0 t On adding U11 and U12 we get U1 =



     φ−1/2 π(m 2 + n 2 + p 2 ) + φ0 π(m 2 + n 2 + p 2 ) − 2 +

0

where, of course, the integral is divergent.

1

1 dt, t2 (7.0.8)

228

Electron sums

In a similar fashion U2 can be expressed as the following four-fold integral:  ∞ ∞ ∞ ∞   U2 = t −1/2 exp −π(x 2 + y 2 + z 2 )t dt d x dy dz. (7.0.9) 0

0

0

0

Changing to polar coordinates and performing the angular integrals leads to  ∞ ∞  ∞  1 dt dt U2 = 4π t −1/2 exp(−π tr 2 )r 2 dr dt = = 1 + , 2 2 t 0 0 0 0 t (7.0.10) where the integral is again divergent. Subtract (7.0.10) from (7.0.8) and ‘cancel’ the identical divergent integrals; then we get, for the simple cubic lattice,      φ−1/2 π(m 2 + n 2 + p 2 ) +φ0 π(m 2 + n 2 + p 2 ) −3. (7.0.11) U (SC) = Tables of φk (x) were prepared by Born and Misra [4]. One only needs a few terms of the series, since φk (x) rapidly becomes smaller as x increases. Thus Coldwell-Horsfall and Maradudin [10] found that U (SC) = −2.837297

e2 . a0

 However, if we look back to Section 1.3, where a(2s) =  1/(m 2 + n 2 + p 2 )2s was treated as the three-dimensional analogue of the Riemann zeta function, we see that U1 (SC) = a(1) and, by making use of the functional equation coming from analytic continuation, a(1) may be evaluated and is given in Table 1.5. To much surprise it was seen that U1 (SC) found for a(1) by the simple method described in Section 1.3 is equal to the expression U (SC) found by the rather complex method given above. Results for electrons forming face-centred cubic (FCC) and body-centred cubic (BCC) lattices are also found in [10]. It is simple to show that for these lattices the interaction amongst the electrons alone is U1 (FCC) = a(1) + d(1),

U1 (BCC) =

1 2

[a(1) + 3c(1)] ,

and it is equally simple to evaluate them; once again they were found to be equal to the values in [10]. That is, e2 e2 U1 (FCC) = U (FCC) = −4.584875 , U1 (BCC) = U (BCC) = −3.639240 . a0 a0 To find the most stable of these lattices the results have to be expressed in terms of rs , the radius of a sphere of volume equal to the volume per electron; we have a03 = 43 πrs3 (SC),

1 3 4 a0

and thus U (SC) = −1.760119

= 43 πrs3 (FCC), e2 , rs

1 3 2 a0

= 43 πrs3 (BCC),

U (FCC) = −1.791753

U (BCC) = −1.791860

e2 , rs

e2 , rs

Electron sums

229

in which the BCC lattice appears as the most stable. Results obtained by Bonsall and Maradudin [3] by the traditional method for two-dimensional electron lattices also agreed with closed-form results found by direct evaluation of just the appropriate U1 . Thus for the square and triangular lattices (Bonsall and Maradudin call the latter the hexagonal lattice) we have   U1 (sq) = (m 2 + n 2 )−1/2 , U1 (tri) = (m 2 + mn + n 2 )−1/2 . A standard decomposition of these double sums into products of single sums, to be found in Zucker and Robertson [19] and Borwein and Borwein [8], gives U1 (sq) = 4ζ ( 12 )L −4 ( 12 ) = −3.900 264 924, U1 (tri) = 6ζ ( 12 )L −3 ( 12 ) = −4.213 422 363 in terms of a0 , the side of the square or of the triangle, and it is very simple to calculate these quantities. In terms of rs , the radius of a circle equal in area to the area per electron, we have a02 = πrs2 (sq),

√ 3 2 2 a0

= πrs2 (tri),

so that U (sq) =

−2.200488843e2 , rs

U (tri) =

−2.212205173e2 . rs

There were two surprises here. First these numbers could be obtained without going through the long process just described, in particular, avoiding subtracting two infinite integrals. Secondly, why was the actual energy of the lattice given just by the interaction of the electrons alone, calculated in this way? Since it is much easier to evaluate the analytic continuations of the corresponding lattice sums without bothering to subtract divergent integrals ‘in the right way’, the question arises whether using this method is valid for all Coulomb lattices. An attempt to justify this procedure was explored by Borwein et al. [6]. They started by observing that the previous manipulations were not arbitrary, but rather were stable since any answer might be obtained by inappropriate processes. Some regular limiting procedure must be undertaken to guarantee a robust answer. The following principle is proposed: the rearrangements used should depend only on the geometry of the underlying lattice and not on the power s in the law of interaction. The consequence of this principle is that we look for an appropriate analytic function for U (lattice : s) and take our answer the value of this function at s = 12 . They argue that this forces the answer to be alat (1) where alat (2s) is the d-dimensional sum over the lattice sites:  (l12 + l22 + · · · + ld2 )−2s , alat (2s) = in which the sum represents alat (2s) for Re s > 12 d. This has an analytic continuation for 0 < Re s < 12 d, which, for the d-dimensional simple cubic lattice, is

230

Electron sums

obtained via the functional relation π −s (s)alat (2s) = π s−d/2 



 1 d − s alat (d − 2s). 2

Formally, U (lattice : s) = alat (2s) − U2 (s), where   d x1 · · · d xd ... U2 = c % 2 &s , x1 + x22 . . . + xd2 () while c is a constant appropriate to the lattice and  is some volume in d-dimensional space which gives electrical neutrality. Now split U2 into the regions inside the unit sphere and outside the unit sphere thus: U2 (s) = U2< (s) + U2> (s). Adding on the finite integral U2< (s) when 0 < Re s < 12 d to both sides, we have U (lattice : s) + U2< (s) = alat (2s) − U2> (s).

(7.0.12)

Now, U2< (s) may be integrated to give for Re s > 12 d U2< (s) =

C d − 2s

with

C=

 c 2π d/2 ( 12 d)

,

whereas for Re s > d/2 the other integral, U2> , is finite and, by direct computation or by appealing to the central symmetry of the integral, has the value C/(2s − d). Now we argue as follows. The left-hand side of (7.0.12), namely U (lattice : s) + C/(d − 2s), is an analytic continuation of the right-hand side of (7.0.12) for Re s < 12 d. Thus, by comparing the two we have U (lattice : s) = alat (2s) for Re s < 12 d and hence for s = 12 we have our result. By considering a precise limiting process we shall demonstrate why a unique ‘answer’ is to be obtained. The following model in d dimensions will be considered. In this model point charges are located at lattice sites and these are surrounded by an equal amount of opposite charge uniformly distributed over hypercubes centred at the lattice point and of side equal to the lattice spacing. This is illustrated there after in two dimensions, where the shaded portion represents positive charge of value equal to the point negative charge but uniformly distributed over a square.

Electron sums

231

We shall thus examine the precise limiting procedure  N N  −s  lim σ N (s) = lim l12 + l22 + · · · + ld2 ···

N →∞

N →∞

 −

−N N +1/2

−N +1/2

···

−N  N +1/2



−N +1/2

x12 + x22 + · · · + xd2

−s

 d x1 d x2 · · · d xd

= lim [α N (s) − β N (s)] , N →∞

although a priori this limit need not exist. This procedure maintains electrical neutrality throughout the limiting process. Further the model has zero dipole moment and, in three dimensions, zero quadrupole moment as well. Consider first the multidimensional integral β N (s) . Writing x j = (N + 12 )X j , we have  1  1  β N (s) 2 2 X = I (s) = · · · + · · · + X d d X 1 · · · d X d . (7.0.13) 1 (N + 12 )d−2s −1 −1 Now, I (s) may be rewritten as follows:   d X1 · · · d Xd I (s) = 2d · · · , 2 2 s Wd (X 1 + · · · + X d ) where Wd is the pyramidal region |X i | ≤ X d ≤ 1. Making the substitution X i = X d xi , we have  1  1  1 d x1 · · · d xd−1 2d d−1−2s C(s), I (s) = 2d Xd d Xd ··· = 2 2 s d − 2s 0 −1 −1 (1 + x 1 + · · · x d−1 ) where

 C(s) =

1 −1

 ···

1

−1

d x1 · · · d xd−1 , 2 )s (1 + x12 + · · · xd−1

and this integral clearly converges for Re s > 0. Thus σ N (s) = α N (s) − (N + 12 )d−2s

2d C(s), d − 2s

(7.0.14)

and the whole of the right-hand side of (7.0.14) is meromorphic for Re s > 0, with a single simple pole at s = 12 d. Note that β0 (s) = 22s−d

2dC(s) (d − 2s)

(7.0.15)

gives an analytic continuation of the integral inside the unit hypercube. For Re s > 12 d, σ N (s) is infinite since its defining integral is infinite. Then, for Re s > 12 d, lim σ N (s) + β0 (s) = lim α N (s) + β0 (s) = alat (2s) + β0 (s),

N →∞

N →∞

(7.0.16)

232

Electron sums

since β N (s) tends to zero. We can use (7.0.15) to see that alat (2s) + β0 (s) is analytic in Re s > 0. The principle of analytic continuation allows us to conclude that (7.0.16) continues to hold in any half-plane in which the left-hand side exists and is known to be analytic. However, for 0 < Re s < 12 d, β0 (s) is a finite integral. Thus, in the appropriate strip, lim N →∞ σ N (s) = a1at (2s) for the particular limiting process we have undertaken – if it is, in fact, true that lim N →∞ σ N (s) + β0 (s) exists and is analytic in the appropriate half-plane (or at least for 12 < Re s < 12 d with continuity at 12 ). Now, for all lattices in two dimensions, σ N (s) + β0 (s) can be shown to tend to an analytic limit in the right half-plane. This was proved for the square lattice in Borwein et al. [6]. While it might seem reasonable to presume that it holds generally, that is in fact false. Considerations similar to those given there show that, for the simple cubic lattice in three dimensions, the limit is analytic for 12 < Re s < 32 but is discontinuous at 12 . Indeed lims→1/2+ lim N →∞ σ N (s) = alat (2s), the correct answer, but this differs by π/6 from lim N →∞ σ N ( 12 ) (the rectangular limit). This is discussed more fully in a further paper by Borwein et al. [5]. In addition, in two dimensions one can also show that the limit over expanding circles, namely the limit as N → ∞ of    (m 2 + n 2 )−s − (x 2 + y 2 )−s d x d y τ N (s) = 1≤(m 2 +n 2 )≤N

1≤(m 2 +n 2 )≤N

is analytic for Re s > 13 . As a consequence, when the integral inside the unit circle is reintroduced, the corresponding limiting value at s = 12 is also 4ζ ( 12 )L −4 ( 12 ). Granted the general applicability of the previous meta-principle – which we have illustrated already for SC, FCC, BCC, square, and triangular lamina lattices – we can easily determine U (lattice) for many other lattices. Thus other threedimensional lattices were investigated; in particular, various types of hexagonal lattice were considered. A simple hexagonal lattice is a structure formed by stacking planes of two-dimensional triangular lattices directly above each other. The direction of stacking is known as the c axis and the separation of planes in terms of the nearest neighbour distance, R, in the triangular lattice is called the axial ratio, c. If particles on such a lattice interact with an r −s potential, it is simple to show that the appropriate lattice sums are given by −s   (m − 12 )2 + 3(n − 12 )2 + c2 p 2 (m 2 +3n 2 +c2 p 2 )−s + . H (2s : c) = Using our principle that the energy of a such a lattice of electrons in a compensating positive background is given in terms of e2 /R by H (hex) = H (1 : c), methods of converting such sums into rapidly converging cosech sums have been previously described in Section 1.3 and, after some detailed algebra, H (1 : c) can be converted to rapidly converging sums and double sums of cosech functions:

Electron sums

233

H (1 : c)

√ √  ∞ 16 log 2 3π c 3  cosech(cπ m) − + 2 =− 3c 18 3 m 1  √ √  ∞ √ ∞ ∞    3 cosech(cπ m/ 3) 3 cosech cπ m 2 + n 2 /3 +2 +4 √ m 3m 2 + n 2 1 1 1

 √ ∞ ∞  ∞   4 3 cosech(2π m/c) 4 cosech π 4m 2 /3 + 4n 2 /c2  + 2 +4 3c 2m/c 4m 2 /3 + 4n 2 /c2 1 1 1 √ √ ∞  cosech( 12π m/c) 4 3 + 2 √ c 12m/c 1

 ∞  ∞  4 cosech π 12m 2 /3 + 12n 2 /c2  +4 12m 2 + 12n 2 /c2 1 1 √  ∞ √ 3  (−1)m cosech( 3π m/c) − 2 √ c 3m/c 1  ∞ ∞   (−1)m+n cosech π (3m 2 /3 + 3n 2 /c2 )   . (7.0.17) +4 3m 2 + 3n 2 /c2 1 1

Evaluations of H (1 : c) for several values of c are given below. The only corresponding results we'know in the literature are those given by Hund [15,

16] for c = 2 and c = 83 . The latter value for c is the so-called ideal ratio. Hund’s values were calculated by the traditional Ewald method and the numerical accuracy was low. He gave  '  H (1 : 2) = −1.796. H 1 : 83 = −2.238, As in the case of the three cubic lattices, we convert the values in terms of e2 /R into values in of e2 /rs . For the simple hexagonal lattice the volume of a √terms 3 unit √ cell is 3 3c R /2, and with three electrons per cell the volume per electron is 3c R 3 /2. Hence √ √ 3c R 3 /2 = 4πrs3 /3 and rs /R = (3 3c/8π )1/3 . √ Multiplying H (1 : c) by (3 3c/8π )1/3 yields U (hex) in terms of e2 /rs , as in Table 7.1. It is so simple to evaluate H (1 : c) via (7.0.17) that numerically the minimum was located at c = 0.9284 with U (hex) = −1.774642655. e2 /rs We have also evaluated U for the hexagonal close-packed (HCP) structure, which may be regarded as two equal interpenetrating hexagonal crystals with ideal

234

Electron sums Table 7.1 Values for U (hex) c2

U (hex)/(e2 /R)

U (hex)/(e2 /rs )

1 4 9 16 3 4

−3.263230504

−1.531505030

−3.253618321

−1.747971578

−3.143633242 −2.995711953 −2.730799747 −2.502936140 −2.238722127 −1.795017494

−1.771826683 −1.771389474 −1.727636634 −1.661251716 −1.558866728 −1.337291317

1 3 2

2 8 3

4

axial ratios, one lattice based at the origin (0, 0, 0) and the other at lattice sums for such a structure in terms of R may be written  '  HCP(2s) = H 2s : 83 + H ∗ (2s),



1 1 1 2, 3, 2

 . The

where H ∗ (2s) =

−s  (m − 12 )2 + 3(n − 13 )2 + 83 ( p − 12 )2 −s  + m 2 + 3(n − 16 )2 + 83 ( p − 12 )2 .

√ For the HCP structure the volume of a hexagonal cell is 3 2R 3 , but there are six electrons per cell. Hence we have √ 3 √ 2R /2 = 4πrs3 /3 and rs /R = (3 2/8π )1/3 , and we obtain U (HCP) = −3.241858662e2 /R = −1.791676267e2 /rs . Foldy [12] found by the traditional method that U (HCP) = −1.79167624e2 /rs and noted that if the axial ratio is 1.0016 times the ideal value then the last two figures read 90 instead of 24, and he claims this is the real minimum. As a final example, the result for the diamond lattice is given. The diamond lattice may be considered as two equal interpenetrating FCC lattices, one based on (0, 0, 0) and the other on ( 14 , 14 , 14 ). In terms of the cube side of an FCC lattice, the diamond lattice sums are easily shown to be di(2s) = FCC(2s) + 22s−1 [BCC(2s) − SC(2s)] ,

Electron sums

235

and again the principle gives U (di) = di(l) in terms of e2 /a0 . Using previous results, di(1) = 12 [a(1) + 3c(1) + 2d(1)] = −5.386789045. For the diamond lattice, the volume of a cubic cell is U and this has eight electrons per cell. Hence a03 /8 = 4πrs3 /3 and rs /a0 = (3/32π )1/3 , and we obtain U (di) = −1.670851406e/rs , in complete agreement with Foldy’s 1978 value, obtained conventionally. It should, however, be pointed out that in all the previous calculations the structures were either Bravais lattices or contained a basis of energy equivalent sites. With more complex lattices where this is not the case, ignoring this point has led to incorrect results for the fluorite, perovskite, and spinel lattices – see Zucker [18]. This was noted by Cockayne [9], who found the correct results by the conventional Ewald approach. Baldereschi et al. [2] showed that by correctly superimposing the component Bravais lattices with the right weightings, the method used here gives the correct values. Thus the perovskite lattice is a simple cubic lattice with a five-fold basis, with the general formula AB X 3 . The A sites are the corners of the cube, the B sites the centres of the cube and the X sites the faces of the cube. The X sites, though energetically equivalent to each other, are not equivalent to the A and B sites. One has to evaluate the interaction of each site with the others and weight them accordingly. Thus the perovskite lattice may be taken to be made up of the A lattice, an SC lattice based at the origin (0, 0, 0), the B lattice, an SC lattice based on ( 12 , 12 , 12 ), and three X lattices, i.e., SC lattices based on the equivalent sites (0, 12 , 12 ), ( 12 , 0, 12 ), and ( 12 , 12 , 0). So, the interaction of A with the other sites may be written, in the notation of Section 1.3, as ψ A = ψ(0, 0, 0) + ψ( 12 , 12 , 12 ) + 3ψ(0, 12 , 12 ) = a(1) + 12 [3c(1) − a(1)] + d(1). Similarly, we have ψ B = ψ(0, 0, 0) + ψ( 12 , 12 , 12 ) + 3ψ(0, 0, 12 ) = a(1) + 12 [3c(1) − a(1)] + 12 [3b(1) − d(1)], and ψ X = ψ(0, 0, 0) + 3ψ(0, 12 , 12 ) + ψ(0, 0, 12 ) = a(1) + f (1) + 16 [3b(1) − d(1)]. To obtain the correct electronic interaction we require to weight these terms correctly: UP =

1 5

(ψ A + ψ B + 3ψ X ) =

1 5

  e2 7b(1) + 7c(1) + 13 13d(1) = 4.671242310 . a0

236

Electron sums

Since there are five electrons per cube, in terms of rs this becomes 1.694 648 083e2 /rs , in complete agreement with Cockayne’s result. Similarly, for the fluorite lattice AB2 , A is an FCC lattice based on (0, 0, 0) and the two B lattices are FCC lattices based on the equivalent sites ( 14 , 14 , 14 ) and ( 34 , 34 , 34 ).As before,   ψ A = ψ FCC (0, 0, 0) + 2ψ FCC 14 , 14 , 14 = a(1) + d(1) + 3c(1) − a(1) and ψ B = ψ FCC (0, 0, 0) + ψ FCC



1 1 1 4, 4, 4



+ ψ FCC



1 1 1 2, 2, 2



= a(1) + d(1) + 12 [3c(1) − a(1)] + a(1) − d(1). Hence, after some further manipulation, we obtain UFluorite =

1 12

(4ψ A + 8ψ B ) =

1 3

2

[3b(1) + 9c(1) + 2d(1)] = 6.380598623 ae 0 .

Since there are 12 electrons per cube, in terms of rs this becomes 1.728906369e2 /rs , again in complete agreement with Cockayne’s result. Of all the three-dimensional structures considered, the BCC remains energetically the most stable.

7.1 Commentary: Wigner sums as limits We now describe an analysis from [5], as follows. The authors investigated the limit as N → ∞ of the d-dimensional quantity σ N (s) := α N (s) − β N (s), where α N (s) = β N (s) =

N 

···

N 

−N −N  N +1/2 −(N +1/2

s f (n 1 , n 2 , . . . , n d ) ,

···



N +1/2

−(N +1/2



f (x1 , x2 , . . . , xd )

(7.1.1)

s

d x1 · · · d xd

for various particular functions f . Two instances were studied in detail. (1) First, the two-dimensional case where f is given by a positive definite binary quadratic form Q(x, y) := ax 2 + bx y + cy 2 was analyzed in [5, Theorem 1]. Namely, for any positive definite form Q, the quantity σ (s) := lim N →∞ σ N (s) exists in the strip 0 < Re s < 1 and coincides therein with the analytic continuation of α(s). Thus, the integral β N (s) plays no role in the final answer. This is very tidy for two-dimensional lattices.

7.2 Commentary: Sums related to the Poisson equation

237

(2) The authors of [5] then investigated the three-dimensional case for the simple cubic lattice, namely Q(x, y, z) := x 2 + y 2 + z 2 and came to a similar conclusion as before, for the strip 12 < Re s < 32 . However, at s = 12 , σ (s) is discontinuous and     σ 12 = α 12 + 16 π. (7.1.2) The physicist’s method of finding lim N →∞ σ N ( 12 ) always alights on α( 12 ); why this is so is not fully understood but some heuristic explanations– regarding analytic continuations – are to be found in [6]. We leave an open question: Can a similar analysis be done for the four-dimensional simple cubic lattice? Presumably, there is a strip for which σ (s) = α(s) but presumably s = 12 will not lie within this strip? In four dimensions the closed form is known for α in the simple cubic case:   α(s) = 8 1 − 22−2s ζ (s)ζ (s − 1); see [8, (9.2.5)].

7.2 Commentary: Sums related to the Poisson equation In a recent treatment of ‘natural’ Madelung constants [11], it is pointed out that the Poisson equation for an n-dimensional point-charge source, ∇ 2 n (r) = −δ(r),

(7.2.1)

gives rise to an electrostatic potential – we call it the bare-charge potential – of the form ( n2 − 1) 1 Cn =: n−2 4π n/2 r n−2 r 1 log r =: C2 log r, 2 (r) = − 2π

n (r) =

if n = 2,

(7.2.2) (7.2.3)

where r := |r|. Since this Poisson solution generally behaves as r 2−n , [11] defines a ‘natural’ Madelung constant Nn as (here, m := |m|): 

Nn := Cn

 (−1)1·m if n = 2, m n−2 n

m∈Z 

N2 := C2

 m ∈ Zn

(−1)1·m log m,

(7.2.4)

238

Electron sums

where, in cases such as this log sum, one must infer an analytic continuation [11] as the literal sum is quite non-convergent. This Nn coincides with the classical Madelung constant 

 (−1)1·m Mn := m n

(7.2.5)

m∈Z

1 only for n = 3 dimensions, in which case N3 = 4π M3 . In all other dimensions there is no obvious M, N relation. A method for gleaning information about Nn is to contemplate the Poisson equation with a crystal charge source, modelled on NaCl (salt) in the sense of alternating lattice charges:  (−1)1·m δ(m − r). (7.2.6) ∇ 2 φn (r) = − m ∈ Zn

Accordingly – on the basis of the Poisson equation (7.2.1) – solutions φn can be written in terms of the respective bare-charge functions n as  φn (r) = (−1)1·m n (r − m). (7.2.7) m ∈ Zn

7.2.1 Madelung variants We have defined the classical Madelung constants (7.2.5) and the ‘natural’ Madelung constants (7.2.4). Following [11] we define a Madelung potential, now depending on a complex s and spatial point r ∈ Zn :  (−1)1·p , |p − r|s n

Mn (s, r) :=

(7.2.8)

p∈Z

We can write limit formulae for our Madelung variants, first the classical Madelung constant,   1 (7.2.9) Mn := lim Mn (1, r) − r→0 r 

 (−1)1·p = , p n

(7.2.10)

p∈Z

and then the ‘natural’ Madelung constant, Nn := lim [φ(r) − (r )] r→0

(7.2.11)



 (−1)1·p = Cn . p n−2 n p∈Z

(7.2.12)

7.2 Commentary: Sums related to the Poisson equation

239

For small even n, this last sum is evaluable. For example, from [8, (9.2.5)] we have 

 (−1)1·p = (1 − 22−s )(1 − 21−s )ζ (s)ζ (s − 1), p 2s 4

(7.2.13)

p∈Z

which with s = 1 yields N4 = −

1 log 2. π2

Similarly, from [8, Exercise 4b, p. 292] we have 

 (−1)1·p = −16(1 − 24−s )ζ (s)ζ (s − 3), 2s p 8

(7.2.14)

p∈Z

which with s = 3 determines that N8 = −

4 ζ (3). π4

Generally, via the Mellin transform Ms θ42n (q) (see below), values of N2n for small n are similarly susceptible. For instance, if G denotes Catalan’s constant, we have 2G 1 − 3, N6 = − 24π π as in [11]. The more complex value N2 is presented in (7.2.21), below. 7.2.2 Relation between crystal solutions φn and Madelung potentials From (7.2.2), (7.2.7), and (7.2.8) we have the general relation, for dimension n = 2, φn (r) = Cn Mn (n − 2, r).

(7.2.15)

Note that, for the case n = 3, the solution φ3 coincides with the classical Madelung potential M3 (1, r) in the sense that φ3 (r) =

1 M3 (1, r), 4π

because C3 = 1/(4π ). Likewise, the ‘natural’ and classical Madelung constants are related by 4π N3 = M3 . The whole idea of introducing ‘natural’ Madelung constants Nn is that this coincidence of radial powers for φ and M potentials holds only in three dimensions. For example, in n = 5 dimensions, the summands for φ5 and M5 (1, ·) involve radial powers 1/r 3 , as in φ5 (r) =

1 M5 (3, r). 8π 2

240

Electron sums

In [11] it is argued that a solution to (7.2.6) is n 1  k=1 cos π m k rk φn (r) = 2 , π m2 n

(7.2.16)

m∈O

where O denotes the odd integers (including negative odds). These φn give the potential within the appropriate n-dimensional crystal, in that φn vanishes on the surface of the cube [− 12 , 12 ]n , as is required via symmetry within an NaCl-type crystal of any dimension. To render this representation more explicit and efficient, we could write equivalently φn (r) =

2n π2

 m 1 ,...,m n

cos π m 1r1 · · · cos π m n rn . m 21 + · · · + m 2n > 0, odd

It is also useful that – owing to the symmetry arising because the summation indices are odd – we can replace in a cavalier way the cosine product in (7.2.16) with a simple exponential: φn (r) =

1  eiπ m·r . π2 m2 n

(7.2.17)

m∈O

7.2.3 Fast series for φn From previous work [11] we know a computational series  1  sinh[π R(1/2 − |r1 |] n−1 k=1 cos π Rk rk+1 , φn (r) = 2π R cosh (π R/2) n−1

(7.2.18)

R∈O

suitable for any nonzero vector r ∈ [− 12 , 12 ]n . The work [11] also gives an improvement in the case of n = 2 dimensions, namely the following form (see Fig. 7.1) for which the logarithmic singularity at the origin has been siphoned off: φ2 (x, y) =

cosh π x + cos π y 2  cosh π mx cos π my 1 log − . 4π cosh π x − cos π y π m(1 + eπ m ) + m∈O

(7.2.19) These series, (7.2.18) and (7.2.19) are valid, respectively, for r1 , x ∈ [−1, 1]. For clarification, we give here the fast series for n = 3 dimensions:    π 2 + q 2 (1 − 2|x|) cos π py cos πqz p sinh  2 2    φ3 (x, y, z) = .  π p 2 + q 2 cosh π p 2 + q 2 p,q > 0, odd

2

(7.2.20)

7.2 Commentary: Sums related to the Poisson equation

241

7.2.4 Closed form for the ‘natural’ Madelung constant N2 The natural Madelung constant for n = 2 dimensions has also been found, on the basis of (7.2.19) (see [11]), to take the value N2 =

4  3 ( 34 ) 1 log . 4π π3

(7.2.21)

We remind ourselves that this closed form was achieved by contemplating the limiting process r → 0 and hence by Coulomb-singularity removal. The derivation of the above N2 form depends on the relation 4 3

  3 4

π3

√ =

K3/2



2 

√1 2

π 3/4

.

We also record the following numerically effective Mellin transform for n > 2:  ∞ % &

1 1 − θ4n e−π x x n/2−2 d x < 0, (7.2.22) Nn = − 4π 0 where the integral is positive since 0 < θ4 (q) < 1 for 0 ≤ q ≤ 1. From this the large-n behaviour of Nn may be estimated as Nn  −

( 12 n − 1) π n/2



 n n(n − 1) − + · · · , 2 2n/2

(7.2.23)

on making the approximations   θ4 (q) = 1 − 2q + O q 4 and 1 − x n = −n(x − 1) +

& % n(n − 1) (x − 1)2 + O (x − 1)3 2

and then integrating term by term. For instance, from (7.2.22) we compute N100 = −8.6175767047403040779 . . . × 1037 while the asymptotic (7.2.23) gives N100  −8.6175767047403038 . . . × 1037 .

242

Electron sums 7.2.5 Closed forms for the φ2 -potential

Theorem 7.1 It can be proved that

  2 1 log 1 + √ , = 8π 3 √ 1 1 1 φ2 ( 4 , 4 ) = log(1 + 2), 4π √ 1 log(3 + 2 3). φ2 ( 13 , 0) = 8π φ2 ( 13 , 13 )

(7.2.24) (7.2.25) (7.2.26)

Proof Consider, for s > 0, V2 (x, y; s) :=

∞ 

[cos π(2m + 1)x][cos π(2n + 1)y] . [(2m + 1)2 + (2n + 1)2 ]s m,n=−∞

(7.2.27)

This V2 -function is related to φ2 by V2 (x, y; 1) = π 2 φ2 (x, y). Treating it as a general lattice sum [8], we derive (with some difficulty)    V2 13 , 13 ; s =2−1−s −(1 − 2−s )(1 − 32−2s )L 1 (s)L −4 (s)

+ 3(1 + 2−s )L −3 (s)L 12 (s) . (7.2.28) The L-functions in (7.2.28) are various Dirichlet series; L 1 is the Riemann ζ -function. Note that 1 − 32−2s factors as (1 + 31−s )(1 − 31−s ), that lims→1 (1 − 31−s )L 1 (s) = log 3, and that √ √ 3π π 1 L −4 (1) = , L −3 (1) = , L 12 (1) = √ log(2 + 3). (7.2.29) 4 9 3 After gathering everything together we have φ2 ( 13 , 13 )

 √  3+2 3 1 1 1 1 log = 2 V2 ( 3 , 3 , 1) = , 8π 3 π

which is (7.2.24). We find more easily that V2 ( 14 , 14 ; s) = 2

∞

m,n=−∞

(−1)m+n = 21−s L −8 (s)L 8 (s) [(4m − 1)2 + (4n − 1)2 ]s (7.2.30)

is a familiar lattice sum [8]. So, with √ π 1 and L 8 (1) = √ log(1 + 2), L −8 (1) = √ 2 2 2 we derive √ 1 φ2 ( 14 , 14 ) = 4π log(1 + 2), which is (7.2.25). Likewise    V2 13 , 0; s = 2−1−s (1 − 2−s )(1 − 32−2s )L 1 (s)L −4 (s)

+ 3(1 + 2−s )L −3 (s)L 12 (s) ,

(7.2.31)

(7.2.32)

7.2 Commentary: Sums related to the Poisson equation

243

which yields

   √  π √ 1 φ2 0, 13 = log 3(2 + 3)2 = log(3 + 2 3), 16π 8 which is (7.2.26).

Remark: The V2 lattice sum can be given as a fast series:  cos π mx cos π ny (7.2.33) V2 (x, y; s) := (m 2 + n 2 )s m,n ∈ O   23/2−s π s  |u + x| s−1/2 = (−1)u K 1/2−s (π n|u + x|) cos π ny, (s) n + n∈O

u ∈Z

where K ν is a standard modified Bessel function. For s = 1 this series collapses further into π 2 times our series (7.2.19) for the Poisson potential φ2 . Using the integer relation method PSLQ [7] to hunt for results of the form

? (7.2.34) exp π φ2 (x, y) = α, for α algebraic, we may obtain and further simplify many results such as the following. Conjecture 7.1 Empirically,   α + 1/α √ 1 ? log α where = 2 + 1, φ2 14 , 0 = 4π 2 '     √ √ 1 ? log 3 + 2 5 + 2 5 + 2 5 , φ2 15 , 15 = 8π   √ γ + 1/γ 1 ? log γ where = 3 + 1, φ2 16 , 16 = 4π 2   √ τ − 1/τ 1 ? 1 1 log τ where = (2 3 − 3)1/4 , φ2 3 , 6 = 4π 2   √    1+ 2− 2 1 ? φ2 18 , 18 = log , √ 4 4π 2−1 '   √ √ μ + 1/μ 1 ? 1 1 log μ where = 2 + 5 + 5 + 2 5; , 10 = φ2 10 4π 2 ?

the notation = indicates that we have only experimental (i.e., extreme-precision numerical) evidence of an equality. Such hunts are made entirely practicable by (7.2.19). Note that for general x and y we have φ2 (y, x) = φ2 (x, y) = −φ2 (x, 1 − y), so we can restrict searches to 12 > x ≥ y > 0, as illustrated in Fig. 7.1. Indeed computations by Glasser and Crandall given in [11] were precipitated from such experiments and led to the following result.

244

Electron sums 0.5

0.0

–0.5 0.6 0.4 0.2 0.0 –0.5 0.0 0.5

Figure 7.1 High-precision plot of the Monge surface z = φ2 (x, y), via the fast series (7.2.19), showing the logarithmic singularity at the origin (the plot is adapted from [11]). In this plot, x, y range over the 2-cube [− 12 , 12 ]2 ; from symmetry one need know the φ2 surface only over the octant 12 > x ≥ y > 0. We are able to establish closed forms for the heights on this surface above certain rational √ pairs (x, y). As just one example, 1 log(1 + 2) ≈ 0.0701. φ2 ( 14 , 14 ) = 4π

Theorem 7.2 For z :=

π (y + i x), 2

     1 − λ(z)/√2   θ2 (z, q)θ4 (z, q)  1 1   = φ2 (x, y) = log  log  √  , (7.2.35)   2π θ1 (z, q)θ3 (z, q) 4π 1 − 1/[λ(z) 2]  where λ(z) =

θ42 (z, e−π ) θ32 (z, e−π )

=

∞  (1 − 2 cos 2z q 2n−1 + q 4n−2 )2 , (1 + 2 cos 2z q 2n−1 + q 4n−2 )2

(7.2.36)

n=1

with q := e−π . Hence, the presumption that, for rational x, y, (7.2.34) is always algebraic is equivalent to the provable conjecture that, for all z = π2 (y +i x) with x, y rational, λ(z) in (7.2.36) is algebraic, and this is proved in [1, Theorem 10]. How to provide computationally assisted proofs of results such as those of Conjecture 7.1 is also discussed in [1].

References

245

7.2.6 Closed forms for n = 3, 4 dimensions The only nontrivial closed-form evaluation of φ3 of which we are aware is that of Forrester and Glasser [8, 13]. Namely, 4π φ3 (1/6) = M3 (1, 1/6) =

 m ∈ Z3

√ (−1)1·m = 3. |m − 1/6|

References [1] D. H. Bailey, J. M. Borwein, R. E. Crandall, and I. J. Zucker. Lattice sums arising from the Poisson equation. J. Phys. A, 46:115201–115232, 2012. [2] A. Baldereschi, G. Senatore, and I. Oriani. Madelung energy of the Wigner crystal on lattices with non-equivalent sites. Solid State Commum., 81:21–22, 1992. [3] L. Bonsall and A. A. Maradudin. Some static and dynamical properties of a twodimensional Wigner crystal. Phys. Rev. B, 15:1959–1973, 1977. [4] M. Born and R. D. Misra. On the stability of crystal lattices. IV. Proc. Camb. Phil. Soc., 36:466–478, 1940. [5] D. Borwein, J. M. Borwein, and R. Shail. Analysis of certain lattice sums. J. Math. Anal. Appl., 143:126–137, 1989. [6] D. Borwein, J. M. Borwein, R. Shail, and I. J. Zucker. Energy of static electron lattices. J. Phys. A: Math. Gen., 21:1519–1531, 1988. [7] J. M. Borwein and D. H. Bailey. Mathematics by Experiment: Plausible Reasoning in the 21st Century, 2nd edition. A. K. Peters, 2008. [8] J. M. Borwein and P. B. Borwein. Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity. Wiley, New York, 1987. [9] E. Cockayne. Comment on ‘stability of the Wigner electron crystal on the perovskite lattice’. J. Phys. Condens. Matter, 3:8757, 1991. [10] R. A. Coldwell-Horsfall and A. A. Maradudin. Zero point energy of an electron lattice. J. Math. Phys., 1:395–404, 1960. [11] R. E. Crandall. The Poisson equation and ‘natural’ Madelung constants. 2012. Preprint. [12] L. L. Foldy. Electrostatic stability of Wigner and Wigner–Dyson lattices. Phys. Rev. B, 17:4889–4894, 1978. [13] P. J. Forrester and M. L. Glasser. Some new lattice sums including an exact result for the electrostatic potential within the NaCl lattice. J. Phys. A, 15:911–914, 1982. [14] K. Fuchs. A quantum mechanical investigation of the cohesive forces of metallic copper. Proc. Roy. Soc. London A, 151:585–602, 1935. [15] F. Hund. Versuch einer Ableitung der Gittertypen aus der Vorstellung des isotropen polarisierbaren Ions. Z. Phys., 34:833–857, 1925. [16] F. Hund. Vergleich der elektrostatischen Energien einiger Ionengitter. Z. Phys., 94:11–21, 1935.

246

Electron sums

[17] E. P. Wigner. On the interaction of electrons in metals. Phys. Rev., 46:1002–1011, 1934. [18] I. J. Zucker. Stability of the Wigner electron crystal on the perovskite lattice. J. Phys. Condens. Matter, 3:2595–2596, 1991. [19] I. J. Zucker and M. M. Robertson. Systematic approach to the evaluation of  2 2 −s (m,n=0,0) (am + bmn + cn ) . J. Phys. A, 9:1215–1225, 1976.

8 Madelung sums in higher dimensions

Here we make a general study of the convergence properties of lattice sums, involving potentials, of the form occurring in mathematical chemistry and physics. Many specific examples are studied in detail. The prototype is the Madelung constant for NaCl, presuming that one appropriately interprets the summation process.

8.1 Introduction Lattice sums of the form arising in crystalline structures – and defined precisely in the next section – have been subject to intensive study. A very good overview is available in [7] and related research may be followed up in [4–6]. These sums are highly conditional in their convergence and the subject of how best to interpret their convergence is discussed in [3–6] and the references therein. The prototype is the Madelung constant for NaCl: ∞  −∞

(−1)n+m+ p  = −1.74756459 . . . , n 2 + m 2 + p2

presuming that one sums over expanding cubes but not spheres [4]. Since the analytic or numerical evaluations of such sums usually proceed by transform (and renormalization) methods, these issues are often obscured, especially in the physical science literature. As we shall illustrate in this paper, while some general theorems are available the precise study of convergence is a delicate and varied subject. Some of our results are unsurprising, others far from intuitive.

248

Madelung sums in higher dimensions

8.2 Preliminaries and notation We shall suppose throughout that Q(x1 , . . . , xk ) :=

k  k 

αi j xi x j ∈ R[x1 , . . . , xk ]

i=1 j=1

is a positive definite quadratic form with αi j = α ji . For a bounded set C in Rk and a positive real number ν we understand νC to be the set (u 1 , . . . , u k ) ∈ Rk such that (u 1 /ν, . . . , u k /ν) ∈ C and write Cν := νC ∩ (Zk \ (0, . . . , 0)). We shall chiefly be interested in C ⊂ Rk , where (0, . . . ,0) lies in the interior of C, so that lim νC = Rk .

ν→∞

We define the corresponding lattice sum Aν (s) = Aν (C, Q, s) :=

 (x1 ,...,xk )∈Cν

(−1)x1 +···+xk Q(x1 , . . . , xk )s

and write A(s) = A(C, Q, s) := lim Aν (C, Q, s) ν→∞

whenever this limit exists. For the most part we shall suppress explicit reference to parameters such as C and Q and simply write A(s) (except in Sections 8.4.2 and 8.7, where we use A(C, Q, s) to emphasize the dependence upon the region C). Throughout, we avoid summing over the pole at zero. Also we often write σ for Re s. (The literature is split as to whether to write A(s) or A(2s); the latter form moves the physically meaningful value from 12 to 1.) At s := 12 our sums are evaluating weighted and signed potentials at the origin over points in the underlying lattice. While we have stated our results with reference to integer lattice points, we x ) by can readily generate analogues for an arbitrary lattice AZk on replacing Q( Q(A x). Notice that a convex body is mapped to a convex body by the matrix A−1 . Our key result is to show that A(s) exists and is analytic at least down to Re s > 12 (k − 1) for all reasonably shaped regions C (and hence that the limit is independent of the shape C chosen in that range). In fact, as the next section shows, the same is true if we replace the factor (−1)x1 +···+xk by a function q(x1 , . . . , xk ) exhibiting a similar degree of cancellation when summed over one of the xi . In Section 8.4 we examine in detail the question of convergence for Re s ≤ 1 (k − 1) when C is an (appropriate) ellipse in Rk , an arbitrary polygon in R2 or 2 R3 with rational vertices, or a k-dimensional rectangle (showing that in the latter

8.3 A convergence theorem for general regions

249

case convergence actually holds for all Re s > 0). In Section 8.5 we give very explicit formulae when Q(x, y) := x 2 + P y 2 for certain P (particularly for P = 3 or 7) and C is the corresponding ellipse x 2 + P y 2 ≤ 1. Several other examples are detailed in Section 8.6. Finally, in Section 8.7, when Q(x, y) := x 2 + y 2 we demonstrate directly the existence and equivalence of the limits at s = 1 (the most analytically pliable value) for C a circle, rectangle, or diamond. Since many of the proofs are lengthy and technical we have chosen to postpone the majority of them until Section 8.8.

8.3 A convergence theorem for general regions Let C be a bounded set in Rk containing (0, . . . , 0) in its interior and let Q(x1 , . . . , xk ) be a positive definite quadratic form in R[x1 , . . . , xk ]. Let q : Zk → R, and let  q(x1 , . . . , xk ) . Aν (s) = Aν (C, Q, q, s) := Q(x1 , . . . , xk )s (x1 ,...,xk )∈Cν

We may now state our basic result. Theorem 8.1

Let χν be the characteristic function of Cν , and suppose that

wν ( j1 , . . . , jk−1 , m) :=

m 

q( j1 , . . . , jk−1 , l)χν ( j1 , . . . , jk−1 , l)

l=−∞

is uniformly bounded for all integers j1 , . . . , jk−1 , m and all positive ν. Then A(s) = A(C, Q, q, s) := lim Aν (s) ν→∞

exists and is analytic in the region σ := Re s > 12 (k − 1). In general, this is the most that we can say, as can be seen by taking C to be (for example) the l1 ball in Rk : Theorem 8.2 Suppose that q(x1 , . . . , xk ) := (−1)x1 +···+xk and that C is the k-dimensional diamond a1 |x1 | + · · · + ak |xk | ≤ c, where a1 , . . . , ak , c ∈ N with d = gcd(a1 , . . . , ak ) and all the ai /d odd; then Aν ( 12 (k − 1)) does not tend to a limit as ν → ∞. Theorem 8.1 shows that in R2 the limit is well defined for σ > 12 , and in R3 for σ > 1, for any sensible region and any reasonable q. We make this precise in the corollary below. We say that a region C in Rk is convex in the ith variable if whenever the points (x1 , . . . , xi , . . . , xk ) and (x1 , . . . , xi , . . . , xk ) are in C then so also is the segment joining them.

250

Madelung sums in higher dimensions

Corollary 8.1 Suppose that q(x1 , . . . , xi , . . . , xk ) is bounded over Zk and is periodic with period M in one variable, xi say, with M 

q(x1 , . . . , xk ) = 0.

xi =1

Suppose further that C is bounded, contains (0, . . . , 0) in its interior, and is convex in the ith variable. Then the conclusions of Theorem 8.1 hold. Indeed, many highly non-convex regions still satisfy Theorem 8.1. We will refer to a vertically convex region in Rk as being one in which the final coordinate exhibits convexity. We now focus on sums over specific regions, showing that in some cases convergence can continue well below σ = 12 (k − 1).

8.4 Specific regions 8.4.1 Lattice sums over sympathetic ellipses k k Given a positive definite quadratic form Q(x1 , . . . xk ) = i=1 j=1 αi j x i x j ∈ Z[x1 , . . . , xk ], with αi j = α ji , and function q : Zk → R we define the arithmetic function  q( x ). r (n, Q, q) := Q( x )=n x∈Zk

In particular when our quadratic form has integer coefficients and we sum over the lattice points in appropriate ellipses Q(x1 , . . . , xk ) ≤ ν, we can replace the k-dimensional lattice sum by a Dirichlet series  q(  r (n, Q, q) x) = Aν (s) = Aν (Q, q, s) := s Q( x) ns Q( x )≤ν x∈Zk \0

1≤n≤ν

and decide when the limit A(s) = A(Q, q, s) := lim

ν→∞

exists by examining the sums S0 (x) = S0 (Q, q, x) :=

 r (n, Q, q) ns

1≤n≤ν



r (n, Q, q).

(8.4.1)

0≤n≤x

We recall the formula (see Hardy and Riesz [8, Theorem 7]) for the abscissa of convergence σ0 > 0 of such a Dirichlet series, σ0 = lim sup x→∞

log |S0 (x)| . log x

(8.4.2)

8.4 Specific regions

251

That is (see [8, Theorem 1]), A(s) will exist for all Re s > σ0 and fail to exist for all Re s < σ0 . We now show that (at least for periodic q with suitable cancellation when summed) convergence over these ellipses always extends below σ = 12 (k − 1). Theorem 8.3 xi . If

Suppose that q(x1 , . . . , xk ) is periodic with period M in each M 

···

r1 =1

M 

q(r1 , . . . , rk ) = 0

rk =1

then the abscissa of convergence σ0 satisfies ⎧ ⎨ 23/73 0 < σ0 ≤ 25/34 ⎩ k/2 − 1

if k = 2, if k = 3, if k ≥ 4.

If M 

···

r1 =1

M 

q(r1 , . . . , rk ) = 0

rk =1

then the abscissa of convergence σ0 = k/2. The theorem follows easily from work done in the past by Landau and Walfisz (for k ≥ 4) and the more recent bounds of Krätzel and Nowak (for k = 3) [11] and Huxley [9] (for k = 2), on the error in approximating the number of lattice points in an ellipsoid by its volume. When k ≥ 4 the bounds in Theorem 8.3 cannot in general be improved. Theorem 8.4 If q(x1 , . . . , xk ) := (−1)x1 +···+xk and Q(x1 , . . . , xk ) := a1 x12 + · · · + ak xk2 , where the ai are all odd positive integers, then for all k ≥ 2 the limit A(s) does not exist for any Re s ≤ 12 k − 1. When k = 2 or 3 one expects the correct upper bounds to be 14 and 12 respectively. From Theorem 8.4, the bound 12 would certainly be sharp when k = 3. In fact we will show that for very general Q and q we have the lower bound σ0 ≥ 14 (k − 1) so that when k = 2 or 3 usually we do indeed have the lower bounds 14 and 12 : Theorem 8.5 M and

Suppose that q(x1 , . . . , xk ) is periodic in all the xi , with period M  r1 =1

···

M  rk =1

q(r1 , . . . , rk ) = 0.

252

Madelung sums in higher dimensions

If r (n, Q, q) (and hence Aν (s)) is not identically 0 then σ0 ≥ 14 (k − 1). Notice that there certainly will be cases with Aν (s) identically zero (with therefore no lower bound on σ0 ); indeed for any M = 2 we can always construct non-trivial periodic q( x ) with q(− x ) = −q( x ) and hence, by symmetry, r (n, Q, q) zero for all n and any Q. For the proof of Theorem 8.5 we will use a technique of Landau to show the existence of a constant c0 = c0 (q, Q) > 0 such that |S0 (x)| > c0 x (k−1)/4 for infinitely many integers x. The method requires some additional notation. k k Given a positive definite quadratic form Q(x1 , . . . , xk ) = i=1 j=1 αi j x i x j in Z[x] with αi j = α ji we let D denote the determinant    α11 . . . α1k     ..  (8.4.3) D :=  ... .    α  k1 . . . αkk and define the positive definite adjoint quadratic form Q ∗ (x1 , . . . , xk ) in Z[x] by Q ∗ (x1 , . . . , xk ) :=

k  k  ∂D xi x j . ∂αi j i=1 j=1

Q ∗∗ ( x)

Notice that = Q( x ) and that when k = 2 we have Q ∗ (x, y) = Q(−y, x). We suppose that q(x1 , . . . , xk ) is periodic in each xi with period M, define the periodic weight function   M M   2π (r1 x1 + · · · + rk xk ) , x ) := ··· q(r1 , . . . , rk ) cos λq ( M r1 =1

rk =1

and set



r ∗ (n) = r ∗ (n, Q, q) :=

λq ( x ).

x∈Zk Q ∗ ( x )=n

For the q( x ) of interest the involved-looking expression for r ∗ (n) often simplifies. For example, when q(x1 , . . . , xk ) := (−1)x1 +···+xs , for some 1 ≤ s ≤ k, we have r ∗ (n) = 2k

 Q ∗ (x1 ,...,xk )=n x1 ,...,xs odd xs+1 ,...,xk even

1.

8.4 Specific regions

253

We remark that r (n, Q, q) is identically zero if and only if r ∗ (n) is identically zero. As with the classical circle problem, our proof relies on the ability to write the sum S0 (x) in terms of Bessel functions Jν (x): √  (  ∞  2π nx ( D M)(k−2)/2 k/4  r ∗ (n) ∗ x , r (n, Q, q) = Jk/2 k/4 M M D n 0≤n≤x

n=1

∗

where denotes that if x is an integer the last term in the sum receives only half weighting: it is 12 r (x, Q, q). In fact we shall actually use integrated forms of this that are more surely convergent. As one consequence of the proof, defining Sν (x) :=

1  (x − n)ν r (n, Q, q) ρ! 0≤n≤x

and setting Bν(ρ) (s) := Aν (s) −

ρ−1  i=0

(s + i) (s)



 Si (ν) q(0) − ν s+i i!

 +

 (s + ρ) q(0) , (s) ρ!

it will be apparent that for any positive integer ρ > 12 (k − 1) we can write   Bν(ρ) (s) = A(ρ) (s) + O ν −(ρ−(k−1)/2+2σ )/2 , where Aρ (s) is analytic in the larger region   , σ > − 12 ρ − 12 (k − 1) + ε, |s| < K , for any fixed K and ε > 0, and possesses the representation √  D M k−2s−1 2k/2+ρ π k/2−1 (s + ρ + 1) ρ A (s) = M (s) 2π  (  ∞ ∗  2π n r (n) ,s Fρ k/2−s M D n

(8.4.4)

n=1

with

 Fρ (z, s) := z 2s−k

∞

v k/2−1−ρ−2s Jk/2+ρ (v) dv.

z

When k = 3 we note some similarity to the relation of Buhler and Crandall [6, (1.5)]. Finally, for k ≥ 4, the work of Novák enables us to extend the optimal bound σ0 = k/2 − 1 of Theorem 8.4 to a broader (if less easily described) class of q and

254

Madelung sums in higher dimensions

Q. We shall say that we are in the non-singular case if, as before, q(x1 , . . . , xk ) is periodic in all xi , with period M and M  r1 =1

···

M 

q(r1 , . . . , rk ) = 0,

rk =1

and, moreover there exist integers h and l > 0 with (h, l) = 1 and lM  r1 =1

···

lM  rk =1



 2πi h Q(r1 , . . . , rk ) = 0. q(r1 , . . . , rk ) exp − l

Note that any such l necessarily has (l, M) = 1. Theorem 8.6 have

For non-singular pairs of q(x1 , . . . , xk ) and Q(x1 , . . . , xk ) we σ0 ≥

k − 1. 2

We shall show in the following corollary that any q(x1 , . . . , xk ) of the form q(x1 , . . . , xk ) := (−1)x1 +···+xs ,

1 ≤ s ≤ k,

is non-singular for all Q(x1 , . . . , xk ), so that we certainly recover Theorem 8.4 by this approach (of course the proof of Theorem 8.6 will be much less elementary than that of Theorem 8.4). Corollary 8.2

If q(x1 , . . . , xk ) is of the form   2πi (a1 x1 + · · · + ak xk ) q(x1 , . . . , xk ) := exp M

for some integers ai then the case is non-singular and σ0 ≥

k −1 2

for all positive definite quadratic forms Q(x1 , . . . , xk ) in Z[x1 , . . . , xk ].   When q(x1 , . . . , xk ) takes the special form exp 2πi M (a1 x 1 + · · · + ak x k ) if xi ≡ bi (mod Mi ) for some integers ai , bi and Mi (with Mi |M) and is zero otherwise, Walfisz [21] showed (see Novák [17]) that in the singular case the upper 1 for k > 4. Thus this division into singular bound can be lowered to σ0 ≤ 14 k − 10 and non-singular cases (although not immediately digestible) is probably the correct characterization as regards the abscissa, and Theorem 8.5 conceivably gives the best general lower bound.

8.4 Specific regions

255

8.4.2 Lattice sums over polygons in R2 and R3 In this subsection we restrict ourselves to the usual weight function q(x1 , . . . , xk ) := (−1)x1 +···+xk . We write A(C, Q, s) for the corresponding lattice sum, rather than merely A(s), to emphasize the dependence here upon the region C (the Q-dependence is of use in the proof of Theorem 8.8 below). For polygons in R2 with rational vertices, we show that either convergence occurs for all Re s > 0 or else convergence fails at s = 12 . Moreover we give an explicit and somewhat surprising diophantine criterion for deciding this, on the basis of the parity of the numerators and denominators of the slopes of the lines making up the perimeters of the polygons. Theorem 8.7 Suppose that P in R2 is a closed polygon with rational vertices whose sides lie on the lines ai x − bi y = ci , where ai , bi , ci ∈ Z with gcd(ai , bi , ci ) = 1 and di = gcd(ai , bi ). (i) If ai /di and bi /di are of opposite parity for all i then A(P, Q, s) exists and is analytic for all Re s > 0. (ii) If ai /di and bi /di are both odd for at least one i then A(P, Q, s) exists for Re s > 12 but does not exist for any real s ≤ 12 . If in addition P is star-shaped around (0, 0) then, even restricting n to an integer, lim An (P, Q, s) does not exist for any real s ≤ 12 . n→∞

We extract two simple cases for further advertisement in the next corollary: Corollary 8.3 (i) If R is a rectangle with rational vertices and sides parallel to the axes then A(R, Q, s) exists for Re s > 0. (ii) If D is the diamond a|x|+b|y| ≤ c, where a, b, c ∈ N with gcd(a, b, c) = 1 and d = gcd(a, b), then A(D, Q, s) exists for Re s > 0 if a/d and b/d are of opposite parity but fails to exist for any real s ≤ 12 if a/d and b/d are both odd. We note that this last corollary allows us to observe that in the Hausdorff metric, or any other reasonable metric, the convex bodies for which convergence works for all σ > 0 are dense in the convex bodies in the unit ball, as are those for which convergence is lost for s = 12 . Theorem 8.6 follows from a more precise version that (for Re s > 0) reduces the problem of convergence over the polygon to the question of convergence solely along the boundary. If we define a variant of the characteristic function

256

Madelung sums in higher dimensions

χC∗ , where points on the boundary of C receive weight 12 , and a corresponding analogue of Aν (P, Q, s), A∗ν (C, Q, s) :=



(−1)x+y

(x,y)∈Cν

∗ (x, y) χνC , Q(x, y)s

then the following is true. Proposition 8.1 Let Q(x, y) = αx 2 +βx y+γ y 2 be a positive definite quadratic form and let P be a closed polygon in R2 with rational vertices whose sides lie on the lines ai x − bi y = ci , where ai , bi , ci ∈ Z with gcd(ai , bi , ci ) = 1. If N is a multiple of all the gcd(ai , bi ) then A∗N (P, Q, s) = F(P, Q, s) + O P,Q (N −2σ ), where F(P, Q, s) is analytic in the whole half-plane σ := Re s > 0. Similarly, in three dimensions we can show that convergence either fails at s = 1 or continues down to s = 12 . Theorem 8.8 Suppose that P is a three-dimensional polygon, containing 0 in its  whose faces lie on the planes ai x + bi y + ci z = interior and star-shaped about 0, ei where ai , bi , ci and ei are integers with di = gcd(ai , bi , ci ). (i) If every face has at least one of ai /di , bi /di , ci /di even, then A(P, Q, s) exists for all Re s > 12 . (ii) If there is at least one face with ai /di , bi /di , ci /di all odd, then A(P, Q, s) exists for all Re s > 1 but fails to exist for any real s ≤ 1. We presume that these two- and three-dimensional theorems can be generalized to more dimensions. We have already seen in Theorem 8.2 that the result on diamonds extends naturally to higher dimensions. In the next subsection we show that the behaviour over squares can be similarly recaptured in arbitrary dimensions. 8.4.3 Sums over rectangles When we sum over k-dimensional rectangles we are able to show that, in general, convergence holds for all Re s > 0. More precisely, given an m  = (m 1 , . . . , m k ) in Nk we define the lattice sum over the corresponding rectangle: Am (s) :=

m1  n 1 =−m 1

···

mk  n k =−m k

q(n 1 , . . . , n k ) Q(n 1 , . . . , n k )s

8.5 Some analytic continuations

257

(where as usual the pole (n 1 , . . . , n k ) = (0, . . . , 0) is omitted) and set A(s) :=

lim

min m i →∞

Am (s)

whenever that limit exists. Theorem 8.9

If the sums l1  n 1 =− j1

···

lk 

q(n 1 , . . . , n k )

n k =− jk

are uniformly bounded for all integers j1 , . . . , jk , l1 , . . . , lk then the limit A(s) exists and is analytic for all σ > 0. 8.4.4 l p -balls: an open question It is natural to make an examination of l p -sums for p ∈ N, p > 2, that is, when k = 2 for example, C := {(x, y) : |x| p + |y| p ≤ 1}. We are able to state (see [10]) asymptotically sharp expressions for the number of lattice points in these regions: 

1=

|n| p +|m| p ≤x

2 2 ( p −1 ) 2/ p 2 x + O(x 1/ p−1/ p ), p(2 p −1 )

p ≥ 3.

(n,m)∈Z2 \(0,0)

Unfortunately the l p ball and the underlying ellipse seem highly unsympathetic to analysis, and we leave as an open question what one can provide in the way of lower or upper bounds on σ0 in this case (the most natural example to consider being p = 4).

8.5 Some analytic continuations We shall write α(s) for the alternating zeta function (also known as the Dirichlet eta function), α(s) :=

∞  (−1)n+1 n=1

ns

,

and L ±d (s) for the L-series L ±d (s) :=

∞ 

(±d|n)n −s ,

n=1

where (d|n) is the Kronecker (generalized Legendre) symbol.

258

Madelung sums in higher dimensions

When Q(x, y) := x 2 + py 2 with p = 3 or 7 we can write down an analytic continuation of our lattice sum in terms α(s) and L − p (s):  (x, y) ∈ Z2 \ (0, 0)



(x, y) ∈ Z2 \ (0, 0)

(−1)x+y = −2(1 + 21−s )α(s)L −3 (s), (x 2 + 3y 2 )s (−1)x+y = −2α(s)L −7 (s), (x 2 + 7y 2 )s

resembling the representation (see Glasser and Zucker [7])  (x, y) ∈ Z2 \ (0, 0)

(−1)x+y = −4α(s)L −4 (s). (x 2 + y 2 )s

These arise from our ability to write  (x, y) ∈ Z2 \ (0, 0)

for the quadratic form

1 = u ζ (s)L − p (s) Q p (x, y)s

(8.5.1)

     p+1 p+1 p−1 x2 + xy + y2 4 2 4 √ when p = 3, 7, 11, 19, 43, 67, or 163 (the primes for which Q( − p) is a unique factorization domain), where u = 6 if p = 3 and 2 otherwise (the number of √ units in Q( − p)). Unfortunately it is not clear how to insert a factor (−1)x+y into (8.5.1), or how to replace the Q p (x, y) in those sums by x 2 + py 2 , other than when p = 3 or 7. Many sums of this type have been obtained by Glasser, Zucker, and Robertson [7, 24, 25] for forms whose discriminant is disjoint (i.e., have one form per genus): 

Q p (x, y) :=

 (x, y) ∈ Z2 \ (0, 0)



(x, y) ∈ Z2 \ (0, 0)

 (−1)x+y 1−t = −2 (1 − (2|μ)21−s )L ±μ L ∓4P/μ , (x 2 + P y 2 )s μ|P

(−1)x (x 2 + 2P y 2 )s

 = −21−t (1 − (2|μ)21−s )L ±μ L ∓8P/μ , μ|P

where L ±μ is taken such that μ ≡ ±1 (mod 4) and where the P’s are certain square-free numbers (≡ 1 (mod 4) in the second case) with t prime factors, the appropriate P’s less than 10000 being P = 5, 13, 21, 33, 37, 57, 85, 93, 105, 133, 165, 177, 253, 273, 345, 357, 385, 1365 and P = 1, 3, 5, 11, 15, 21, 29, 35, 39, 51, 65, 95, 105, 165, 231 respectively (in the latter case similar representations can be shown to hold for x 2 + 8P y 2 ). Zucker and Robertson also obtained results for the forms x 2 + P y 2 , x 2 + 4P y 2 , and x 2 + 16P y 2 when P = 3, 7, or 15

8.6 Some specific sums

259

(thus including the continuations of our sums above, although their approach is different from ours).

8.6 Some specific sums We have now obtained from the above theorems very explicit if quite contrasting results regarding the range of convergence for shapes such as circles, diamonds and squares. We continue with some related examples. (1) It was shown in Borwein, Borwein, and Taylor [4, Section VI] that the study of the Madelung constant for a two-dimensional hexagonal lattice with ions of alternating unit√ charge placed at the lattice points of a lattice with basis vectors (1, 0) and ( 12 , 23 ) leads naturally to sums of the form  (n,m)∈C N

(n 2

q(n, m) , + nm + m 2 )s

where q(n, m) :=

4 3

sin[ 23 (n + 1)π ] sin[ 23 (m + 1)π ] −

4 3

sin 23 nπ sin[ 23 (m − 1)π ].

Applying the above theorems, it is clear, on splitting q up appropriately, that when Re s > 12 convergence occurs in such a sum when the lattice points are summed over any vertically convex set, that for expanding rectangles convergence holds for all Re s > 0, and that, when summing over expanding ellipses m 2 + mn + n 2 ≤ N , convergence fails at some point between σ = 14 and σ = 23 73 . Notice that (in the notation of Section 8.4.1)  1. r ∗ (n) = 92 x 2 +x y+y 2 =n x≡y≡0(mod 3)

This sum was also shown to possess a similar analytic continuation to those mentioned in Section 8.5: h 2 (s) = 3(1 − 31−s )(1 − 21−s )−1 α(s)L −3 (s). (2) In [7], sums like c2 (s) :=

 ( j,k)∈Cν

(−1) j ( j 2 + k 2 )s

and c3 (s) :=

 ( j,k, p)∈Cν

(−1) j ( j 2 + k 2 + p 2 )s

260

Madelung sums in higher dimensions

are discussed; again they are covered by our previous analysis, with convergence over vertically convex sets holding for all σ > 12 (respectively 1) and over circles 1 (respectively spheres) but failing at some point between 14 and 23 73 (respectively 2 and 34 ). After a change of variables j = j  + k (respectively j = j  + k + p), convergence over squares (respectively cubes) can be seen (from Section 4.2) to fail at 12 (respectively 1). However (from Section 4.3) convergence does hold for all σ > 0 over certain other parallelepipeds. (3) Let r N (n) denote the number of representations of n as a sum on N squares (counting each permutation and sign change as a separate representation). Then the Dirichlet sum ∞  r N (n) (−1)n s b N (s) := n n=1

is a special case of the sums covered in Theorem 8.3 and Theorem 8.4. In particular, it follows from [5, p. 290] that b4 (s) := −8α(s)α(s − 1). Notice that from our prior analysis the abscissa of convergence is exactly equal to 1. Correspondingly, b3 ( 12 ) is the Madelung constant for sodium chloride. Theorem 8.9 recovers the fact that the limit is taken appropriately if we sum over hypercubes. Theorem 8.2 shows that diamonds fail below 1, and Theorems 8.3 and 8.4 show that the exact abscissa for convergence over spheres lies between 12 and 34 . Theorem 8.4 recaptures the argument in [4] that shows that convergence fails at 12 , and we would conjecture that convergence obtains for σ > 12 .

8.7 Direct analysis at s = 1 In the most basic case, Q(x, y) := x 2 + y 2 and s := 1, one can directly establish that the limit A(C, Q, 1) = −π log 2 when C is the square |x| ≤ 1, |y| ≤ 1, the diamond |x| + |y| ≤ 1, or the circle x 2 + y 2 ≤ 1 (i.e., summing over the standard l p balls for p = 1, 2, or ∞) as we show below. Observe that, when C is the above unit circle, A(C, Q, 1) = lim

n→∞

∞

 00

We show next that the method of expanding diamonds yields the same value. This amounts to proving that 

Tn :=

00

|i|+| j|>0

|i|≤n,| j|≤m

|i|≤n,| j|≤m

We shall prove that Rm,n → −π log 2 when μ := min(m, n) → ∞. Observe that  1 Rm,n = − Rm,n (−t) dt 0

and that, for 0 < t < 1, ⎞ ⎛    1 2 2 2 2 2 2 f (t) − Rm,n (t) = ⎝ ti + j + ti + j + ti + j ⎠ t |i|>n,| j|>m |i|≤n,| j|>m |i|>n,| j|≤m ⎞ ⎛ ∞ ∞ n ∞ ∞ m       4 2 2 2 2 2 2 = ⎝ ti tj + ti tj + ti tj ⎠. t i=n+1 j=m+1

Observe also that, for 0 < t < 1,        ∞ 2 2 i   (−t)  < t (n+1)   i=n+1

i=0

and

j=m+1

i=n+1

  n   2    (−t)i  < 1.   i=0

j=0

264

Madelung sums in higher dimensions

Hence, for 0 < t < 1, | f (−t)−Rm,n (−t)| < 4t (n+1) and so |Rm,n

2 +(m+1)2 −1

+4t (m+1)

2 −1

+4t (n+1)

2 −1

< 12t (μ+1)

2 −1

,

   1  1  12   + π log 2| =  f (−t) dt − Rm,n (−t) dt  ≤ →0  0  (μ + 1)2 0 as μ → ∞.

This completes the proof.

8.8 Proofs Proof of Theorem 8.1

We first note that for a positive definite quadratic form

Q(x1 , . . . , xk ) =

k  k 

αi j xi x j ∈ R[x1 , . . . , xk ],

i=1 j=1

with αi j = α ji , we have Q(x1 , . . . , xk ) ≥ λ(x12 + · · · + xk2 ) for some λ = λ Q > 0. Suppose in what follows that σ > 12 (k − 1), ν > 0, and the ji are integers. Let Q( j1 , . . . jk , s) := Q( j1 , . . . , jk )−s

Q(0, . . . , 0, s) := 0,

when j12 + · · · + jk2 = 0, and let μ :=

sup

ν>0 ( j1 ,..., jk )∈Zk

|wν ( j1 , . . . , jk )| < ∞.

Then we have Aν (s) = =

∞ 

···

∞ 

q( j1 , . . . , jk )χν ( j1 , . . . , jk )Q( j1 , . . . , jk , s)

j1 =−∞ jk =−∞ ∞ ∞  

···

wν ( j1 , . . . , jk )[Q( j1 , . . . , jk−1 , jk , s)

j1 =−∞ jk =−∞

−Q( j1 , . . . , jk−1 , jk + 1, s)] ∞ ∞   =: ··· a j1 ,..., jk (ν, s). j1 =−∞ jk =−∞

8.8 Proofs

265

Suppose next that σ > 12 (k − 1) +  > 12 (k − 1). Observe that, for 0 < u < v,    v      v−u 1 −1−s  − 1  = s t dt  ≤ 1+σ |s|  us vs   u u and hence that, when ( j12 + · · · + jk2 )( j12 + · · · + ( jk + 1)2 ) = 0, |a j1 ,..., jk (ν, s)| ≤ = ≤

|Q( j1 , . . . , jk ) − Q( j1 , . . . , jk + 1)| |s|μ λ1+σ ( j12 + · · · + jk2 )1+σ k−1 αik ji + αkk (2 jk + 1)| |2 i=1 |s|μ λ1+σ ( j12 + · · · + jk2 )1/2 ( j12

λ−1−σ |s|M , + · · · + jk2 )σ +1/2

where M :=

( j12 + · · · + jk2 )σ +1/2

|2

sup

k−1 i=1

αik ji + αkk (2 jk + 1)|

( j12

j12 +···+ jk2 >0

+ · · · + jk2 )1/2

μ < ∞.

Also, when ( j12 + · · · + jk2 )( j12 + · · · + ( jk + 1)2 ) = 0, |a j1 ,..., jk (ν, s)| ≤

μ . |αk,k |σ

Since (as in Borwein, Borwein, and Taylor [4])  ( j2 j12 +···+ jk2 >0 1

1 < ∞, + · · · + jk2 )(k/2)+

it follows, by the Weierstrass M-test, that, when ν → ∞, Aν (s) → A(s), say, uniformly in the region {s : σ > 12 (k − 1) + , |s| ≤ K } with K any fixed positive number and, since Aν (s) is analytic in this region, that A(s) is analytic therein. Consequently A(s) = lim An (s) n→∞

exists and is analytic in the region Re s > 12 (k − 1). Proof of Theorem 8.2 We suppose that C is the diamond a1 |x1 | + · · · + ak |xk | ≤ c where d = gcd(a1 , . . . , ak ) and that all the ai /d are odd positive integers. Observing that when a1 |x1 | + · · · + ak |xk | = cn and n is a multiple of d we have (−1)x1 +···+xk = (−1)cn/d

266

Madelung sums in higher dimensions

and

⎛ |Q(x1 , · · · , xk )| ≤ ⎝

k  k 

⎞ |αi, j |⎠ c2 n 2 ,

i=1 j=1

it is not hard to see that, when s = 12 (k − 1) and N is a multiple of B := a1 · · · ak ,     A N (s) − A N −1/c (s) = Q(x1 , . . . , xk )−(k−1)/2 a1 |x1 |+···+ak |xk |=cN

>

1 c1 N k−1

1 c1 N k−1 > c2 > 0 ≥



1

|x1 |+···+|xk |=(cN /B)



(cN /B) + k − 1 k−1



as N → ∞ (where in fact we have only bothered to count the points with xi = (B/ai )xi ≥ 0); and the limit cannot exist. Proof of Corollary 8.1

Under the hypothesis given, it is simple to estimate that m 

wν ( j1 , . . . , jk−1 , m) :=

q( j1 , . . . , jk−1 , l)χν ( j1 , . . . , jk−1 , l)

l=−∞

is uniformly bounded. Proof of Theorem 8.3 Suppose that q(x1 , . . . , xk ) is periodic with period M; then (with S0 (x) as defined in (8.4.1)), dividing the sum into residue classes modulo M we have S0 (x) =

M  r1 =1

···

M 

q(r1 , . . . , rk )F0 (r1 , . . . , rk , x),

(8.8.1)

rk =1

where F0 (r1 , . . . , rk , x) :=



1.

Q(x1 ,...,xk )≤x xi ≡ri mod M

Writing xi = M yi + ri it is easy to see that F(r1 , . . . , rk , x) counts the number of lattice points (y1 , . . . , yk ) ∈ Zk in the expanded ellipse x 1/2 E, where E is the ellipse M 2 Q(y1 , . . . , yk ) ≤ 1 of area A=

π k/2 √ M k |D| ( 12 k + 1)

with its centre shifted to (−r1 /M, . . . , −rk /M). Approximating the number of points in the ellipse by its area we can write F0 (r1 , . . . , rk , x) = Ax k/2 + O(ψ),

8.8 Proofs

267

where from the results of Huxley [9, Theorem 5] (for k = 2), Krätzel and Nowak [11] (for k = 3), Walfisz [20], and Landau [12, 13] (for k ≥ 8 and k ≥ 4 respectively; see Landau [14, Satz II] for the most immediately applicable form), we can take ⎧ 23/73 x (log x)315/146 if k = 2, ⎪ ⎪ ⎨ 25/34 (log x)10/17 if k = 3, x ψ= 2 ⎪ x if k = 4, x log ⎪ ⎩ k/2−1 if k ≥ 5. x Hence S0 (x) = B1 x k/2 + O (B2 ψ) , where B1 :=

M 

···

r1 =1

M 

q(r1 , . . . , rk ),

B2 :=

rk =1

M 

···

r1 =1

M 

|q(r1 , . . . , rk )|,

rk =1

and the result is plain from (8.4.2). Proof of Theorem 8.4 Notice that if Q(x1 , . . . , xk ) = a1 x12 + · · · + ak xk2 with all the ai odd positive integers and q(x1 , . . . , xk ) = (−1)x1 +···+xk then  1. r (n, Q, q) = (−1)n Q(x1 ,...,xk )=n

Now, by elementary methods we have   |r (n, Q, q)| =

1 = A[1 + o(1)]x k/2

Q(x1 ,...,xk )≤x

n≤x

as x → ∞, where A is the area of the ellipse Q(x1 , . . . , xk ) ≤ 1. In particular there must certainly be infinitely many integers n with |r (n, Q, q)| > 12 An k/2−1 . Hence, if Re s ≤ k/2 − 1 then |A N (s) − A N −1 (s)| =

|r (N , Q, q)| → 0 Nσ

as N → ∞, and the limit A(s) cannot exist. Proof of Theorem 8.5 We closely follow the proof of the corresponding result for the error in the classical circle problem (as given in Landau [15]) and for ρ ≥ 1 inductively define  x Sρ (u) du, Sρ+1 (x) := 0

where S0 (x) is as defined in (8.4.1) (so that equivalently  x 1  1 (x − n)ρ r (n, Q, q) = S0 (u)(x − u)ρ−1 du Sρ (x) = ρ! (ρ − 1)! 0 0≤n≤x

268

Madelung sums in higher dimensions

for all ρ ≥ 1). We invoke the following lemma of Riesz [18] (as in Wilton [23]; cf. Landau [15, Satz 533]). Lemma 8.1 and x > 0,

If f 0 (x) is L-integrable and bounded over (0, x), if, when γ > 0 1 f γ (x) := (γ )



x

f 0 (u)(x − u)γ −1 du,

0

and if V (x) and W (x) are increasing functions of x with | f 0 (x)| < V (x)

and

| fl (x)| < W (x)

then | f β (x)| < b(β, l)V (x)(1−(β/l)) W (x)(β/l) for all 0 ≤ β ≤ l, where the b(β, l) depend only on β and l. We shall show that (as long as Aν (s) is not identically zero) there is a positive integer ν and non-zero constants Bν+1 and Cν such that |Sν+1 (x)| < Bν+1 [1 + o(1)]x (k+2(ν+1)−1)/4 for all x and |Sν (x)| > |Cν |[1 + o(1)]x (k+2ν−1)/4 for infinitely many x. Hence, by the above lemma we must have  ν+1 |Cν | |S0 (x)| ≥ [1 + o(1)] x (k−1)/4 ν/(ν+1) b(ν, ν + 1)Bν+1 for infinitely many x. Thus it remains to justify the claimed upper and lower bounds for the Sρ (x). We define the constants √ ∞  π r ∗ (n) ( D M)(k+2ρ−1)/2  cos(k + 2ρ + 1) , Cρ := 4 Mπ ρ+1 n (k+2ρ+1)/4 n=1 √ ∞  ( D M)(k+2ρ−1)/2  |r ∗ (n)| Bρ := . Mπ ρ+1 n (k+2ρ+1)/4 n=1

Lemma 8.2 We suppose that q(x1 , . . . , xk ) is periodic in all the xi with period M and M M   ··· q(r1 , . . . , rk ) = 0. r1 =1

rk =1

8.8 Proofs

269

For all ρ > 12 (k − 1) we have

  |Sρ (x)| ≤ Bρ 1 + O(x −1/2 ) x (k+2ρ−1)/4

for all x and, if Cρ = 0,

   log x −1/4 |Sρ (x)| = Cρ 1 + O x (k+2ρ−1)/4 log log x

for infinitely many integers x. Proof For ρ ≥ 0 we define inductively r , x) := Fρ+1 (



x 0

Fρ ( r , u) du

and observe (by repeated integration of (8.8.1)) that Sρ (x) =

M 

···

r1 =1

M 

q(r1 , . . . , rk )Fρ (r1 , . . . , rk , x).

rk =1

It has been shown by a number of authors (see for example Landau [12]) that the Fρ ( r , x) can be expressed in terms of Bessel functions: √ ( D M)(k+2ρ−2)/2 r , x) = Vρ (x) + E( r , x), Fρ ( Mπ ρ where π k/2 Vρ (x) := x k/2+ρ M k D ( 12 k + ρ + 1) and E( r , x) := x k/4+ρ/2

 ∞  2π r ∗ (n; r) J k/2+ρ M n k/4+ρ/2 n=1

with ∗

r (n; r) :=

 x∈Zk Q ∗ ( x )=n

(

nx D

 2π r · x . cos M 

Notice that, from the straightforward bounds  1 = O(z k/2 ), Jν (z) = O(z −1/2 ), x∈Zk Q ∗ ( x )≤z

such a sequence is absolutely convergent for ρ > 12 (k − 1). Hence, if M  r1 =1

···

M  rk =1

q(r1 , . . . , rk ) = 0

 ,

270

Madelung sums in higher dimensions

then we obtain

√  (  ∞ 2π nx ( D M)(k+2ρ−2)/2 k/4+ρ/2  r ∗ (n) Sρ (x) = . x Jk/2+ρ ρ k/4+ρ/2 Mπ M D n n=1

Approximating the Bessel functions by cosines (see for example Watson [22, p. 199]), so that (  2 π + O(z −3/2 ), Jν (z) = cos z − (2ν + 1) πz 4 we obtain √ ( D M)(k+2ρ−1)/2 (k+2ρ−1)/4 Sρ = x (M1 + M2 ) Mπ ρ+1 with  (  ∞  π r ∗ (n) 2π nx − (k + 2ρ + 1) cos M1 := (k+2ρ+1)/4 M D 4 n n=1

and

 M2 := O x

−1/2

∞  n=1

(the latter bound follows since 

r ∗ (n)



n (k+2ρ+3)/4

  = O x −1/2

|r ∗ (n)| = O(z k/2 )

(8.8.2)

n≤z

and by assumption 2ρ + 1 > k). The trivial bounding |M1 | ≤

∞  n=1

|r ∗ (n)| n (k+2ρ+1)/4

gives us the required upper bound. By the box principle (in k dimensions), given an N and also k real numbers ν1 , . . . , νk , there is certainly an integer 1 ≤ m ≤ N k + 1 such that the distances from the mνi to their nearest integers simultaneously satisfy ||mνi || < 1/N . √ N In particular √ (taking k = N ) there is an integer m = z ≤ N + 1, with √ m( n/M D) close enough to an integer for n = 1 to N that     ( π π 1 2π nz − (k + 2ρ + 1) = cos(k + 2ρ + 1) + O cos M |D| 4 4 N   log log 16z π = cos (k + 2ρ + 1) + O 4 log 16z for all n = 1, . . . , N ; the latter equality follows from the observation that, since log 16z > e, log 16(N N + 1)2 log 16z ≤ = O(N ). log log 16z log log 16(N N + 1)2

8.8 Proofs

271

Using the estimate (8.8.2) we can readily bound the remaining terms in the sum by ∞  n=N

r ∗ (n) n (k+2ρ+1)/4

= O(N −(2ρ+1−k)/4 ) = O(N

−1/4

 )=O

log log 16z log 16z

1/4  .

Hence, for such a z, as long as Cρ = 0 we have     ∞ r ∗ (n) π log log 16z 1/4 , M1 = cos (k + 2ρ + 1) 1+O (k+2ρ+1)/4 4 log 16z n n=1

and the remaining bound is plain. Varying N (and hence the closeness of the approximation) we can clearly generate infinitely many integers z in this way. Final step of the proof of Theorem 8.5 Hence it only remains to justify that (as long as r (n, Q, q) is not identically zero) Cρ is non-zero for some ρ. First, observe that r ∗ (n) cannot be identically zero. If r ∗ (n) were identically zero then we would have Sρ (x) = 0 for all x ≥ 0 and any ρ > 12 (k − 1); in particular it would follow from the relation r (N , Q, q) = (N + 1)! Sρ (N + 1) −

N −1 

(N + 1 − n)ρ r (n, Q, q)

n=0

and an easy induction on N that r (n, Q, q) would have to be identically zero. We suppose that w is the smallest positive integer such that r ∗ (w) = 0. From the lower bound Q ∗ (x1 , . . . , xk ) > λ∗ (x12 + · · · + xk2 ) we certainly have the trivial lower bound ⎛ ⎞   M M   2 k ∗ k/2 ⎝ ⎠ |r (n)| ≤ Bn , B := ··· |q(r1 , . . . , rk )| . √ λ∗ r =1 r =1 1

Hence if



R ≥ N := max we have

k

7   2  1 π B k k/2+2 log 1 + + 2 , log (w + 1) 2 12 |r ∗ (w)| w

  ∞      ∞ k/2+2    r ∗ (n)   r ∗ (w)  B(w + 1) 1 <    ≤   wR  , R  (w + 1) R n2 n=w+1 n  n=1

 ∗ R and ∞ one of ρ = [(4N − k)/2] + 1 and n=1 r (n)/n  = 0. In particular at least  1 ∗ (k+2ρ+1)/4 and cos[(k + ρ = [ 2 (4N − 1 − k)] + 1 will have both ∞ n=1 r (n)/n 2ρ + 1)π/4] non-zero, and hence Cρ = 0 .

272

Madelung sums in higher dimensions

Proof of the representation (8.4.4) By partial summation and integration by parts we obtain  ν  S0 (ν) − q(0)  d (u −s ) du Aν (s) = − [S0 (u) − q(0)] νs du 1

=

 ρ  (s + i) Si (ν) i=0

=

ρ−1  i=0

(s)

(s + i) (s)

ν s+i 

  (s + ρ + 1) ν − Si (1) + Sρ (u)u −s−ρ−1 du (s) 1

 Si (ν) (s + ρ) Sρ (1) − Si (1) − ν s+i (s)

(s + ρ + 1) + (s)



∞

Sρ (u)u −s−ρ−1 du + O(ν −(ρ−(k−1)/2+2σ )/2 ),

1

for bounded |s|, since by Lemma 8.2 Sρ (u) = O(u (k+2ρ−1)/4 ) for ρ > 12 (k − 1). Proof of Theorem 8.6

Writing

r (n, r) = r (n; r1 , . . . , rk ) :=



1,

Q(x1 ,...,xk )=n xi ≡ri (mod M)

for Re s > 0 we define the theta function ˆ θ(s) :=

∞ 

r (n, Q, q)e−ns

n=0

=

M  r1 =1

···

M 

q(r1 , . . . , rk )θˆ (s; r )

rk =1

where θˆ (s; r ) :=

∞ 

r (n; r )e−ns .

n=0

Now, by Novák [16, Lemma 1] for integers h and l (with (h, l) = 1 and l > 0) the modular functions θˆ (s; r) can be expanded in a neighbourhood of the cusp

8.8 Proofs

273

2πi h/l, so that, for Re s > 0, θˆ (s; r) =

π k/2 (s − 2πi h/l)−k/2 √ D M k/2l k   × Sh,l (m,  r) exp − m  ∈ Zk

where Sh,l (m,  r ) :=

l 

···

a1 =1

l  ak =1

  π 2 Q ∗ (m) , D M 2l 2 (s − 2πi h/l)

  2πi 2πi h Q( a M + r ) + m  · ( a M + r ) . exp − l lM

Thus

    2πi h π k/2  r) + E h,l ( ˆθ σ + Sh,l (0, r, σ) , , r = √ l D M k/2l k σ k/2 where, for fixed h and l,    π 2 λ∗ k 2 2 E h,l ( (m + · · · + m k ) r, σ) ≤ l exp − D M 2l 2 σ 1 k m∈  Z m  =0

=O

∞ 

 n

k/2 −cn/σ

e

= O(e−c/σ ).

n=1

Now, if we are in the non-singular case we can pick h and l such that A := √

M 

π k/2 D M k/2l k

···

r1 =1

M 

 r) = 0 q(r1 , . . . , rk )Sh,l (0,

rk =1

(notice that, since the q(r1 , . . . , rk ) sum to zero, h = 0) and hence   2πi h = A + O(e−c/σ ) σ k/2 θ σ + l uniformly in σ . With S0 (x) as in (8.4.1), writing  ∞ ˆθ (s) = s e−xs S0 (x) d x, 0

for all x would imply that it is clear that |S0 (x)| <    ∞ 2πi h k/2 ˆ k/2 < c |σ + (2πi h/l)| σ σ θ σ+ e−σ x x k/2−1 d x l 0 cx k/2−1

= c |σ + (2πi h/l)| ( 12 k). Hence, on letting σ → 0, we see that, for any constant c1 < A(2π h/l)−1 (( 12 k))−1 , we must have |S0 (x)| > c1 x k/2−1 for infinitely many integers x.

274

Madelung sums in higher dimensions

Proof of Corollary 8.2

We take h = 1 and l to be a high power of M: l := M γ ,

γ > 2α,

where α is the highest power of a prime factor of M dividing 2k D (recall that D, defined in (8.4.3), is the determinant of the matrix of coefficients αi j of Q(x1 , . . . , xk )). Writing R := {(x1 , . . . , xk ) ∈ Zk : 1 ≤ xi ≤ l} and

  2πi Q( x ) q( x ), F( x ) := exp − l

non-singularity will follow once we show the non-vanishing of S := M −k

lM  x1 =1

Now, we have



|S|2 =

lM 

···

F( x) =



F( x ).

x∈R

xk =1

F(y )F( x)

x∈R y∈R



=

F( u + x)F( x)

x∈R u∈R



=

F( u)

u∈R

⎧ k ⎨ l  i=1

⎩

xi =1

⎫ k ⎬ 2πi   xi 2 exp − αi j u j ⎭ l 

j=1

on noting that q( u + x) = q( u )q( x) and making the expansion Q( u + x) − Q( x ) = Q( u) +

k  i=1

⎛ xi ⎝2

k 

⎞ αi j u j ⎠ .

j=1

Observing that l  xi =1

and setting L :=

⎧ ⎨ ⎩

   2πi xi α l = exp − 0 l

(u 1 , . . . , u k ) ∈ R : 2

k  j=1

if α ≡ 0 (mod l), otherwise,

αi j u j ≡ 0 (mod l), 1 ≤ i ≤ k

⎫ ⎬ ⎭

,

8.8 Proofs we obtain |S|2 = l k



275

F( u ).

u∈L

It is not hard to check that any u satisfying the linear system in L must necessarily satisfy 2k Du i ≡ 0 (mod l),

1 ≤ i ≤ k,

u i ≡ 0 (mod M γ −α ),

1 ≤ i ≤ k.

and therefore certainly

Since we have chosen γ > 2α we thus have q( u ) = 1,

Q( u ) ≡ 0 (mod l),

for any u ∈ L, giving |S|2 =



1 = 0

u∈L

(plainly (l, . . . , l) is in L) and we are in the non-singular situation for any Q(x1 , . . . , xk ). Proof of Proposition 8.1 (see Section 8.4) Clearly it is enough to show the result for triangles T , and in fact (by taking sums and differences) enough to consider triangles with one vertex at the origin. Using the symmetries for x → −x, y → −y we shall further assume that the triangle T lies entirely in the quadrant x, y ≥ 0 and (replacing N by cN or N /gcd(a, b) as necessary) that T takes the form T = {(x, y) : r2 x ≥ s2 y, r1 x ≤ s1 y, ax − by ≤ 1}, where a, b, si , ri ∈ Z with ri , si ≥ 0 and gcd(a, b) = gcd(ri , si ) = 1. We denote the sides of T by li l1 : r1 x = s1 y,

l2 : r2 x = s2 y,

l3 : ax − by = 1,

and the points of intersection of l3 with l1 and l2 by P1 and P2 respectively:   si ri . Pi = , asi − bri asi − bri Choosing integers x0 , y0 satisfying ax0 − by0 = 1 and writing Ai = ri x0 − si y0 ,

Bi = asi − bri ,

αi =

Ai Bi

276

Madelung sums in higher dimensions

(notice that gcd(Ai , Bi ) = 1 and Bi > 0), we can parameterize the integer points on n(l3 ∩ T ), the intersection of the line ax − by = n with N T by x = nx0 + bt,

y = ny0 + at,

nα1 ≤ t ≤ nα2

for n = 1 to N , with (−1)x+y = (−1)n(x0 +y0 )+t (a+b) . We distinguish the two cases (i) 2|ab, (ii) 2  |ab. Case (i): a and b are not both odd. Since a and b are of opposite parity we can pick x0 , y0 to both be odd (indeed, either (x0 , y0 ) or (x0 + b, y0 + a) will be of this form). Hence on the line segment n(l3 ∩ T ) our parameterization gives (−1)x+y = (−1)t and, writing f n (t) := Q(nx0 + bt, ny0 + at)−s , we have A N (T, Q, s) =



An (l3 ∩ T, Q, s),

1≤n≤N

where



An (l3 ∩ T, Q, s) =

(−1)t f n (t).

nα1 ≤t≤nα2

We first observe some elementary bounds on f n : | f n (t)| = |Q(x, y)−s | = O(n −2σ ),

| f n (t)| = | − s (2αb + βa)x + (2γ a + βb)y Q(x, y)−s−1 | = O(n −2σ −1 ),   

2 (s + 1) Q(a, b)  | f n (t)| =  (2αb + βa)x + (2γ a + βb)y − 2s Q(x, y)s+2 Q(x, y)s+2  = O(n −2σ −2 ).

(8.8.3)

Pairing odd and even t we have An (l3 ∩ T, Q, s) = M1 + M2 , where M1 :=





f n (2t) − f n (2t + 1)



1 1 2 nα1 ≤t≤ 2 nα2

and M2 := u 2 (n) f n (2tn,2 + 1) − u 1 (n) f n (2tn,1 + 1)   with u 1 (n) := 1 if there is an integer tn,1 in n2BA11 − 12 , n2BA11 and 0 otherwise, and   u 2 (n) := 1 if there is an integer tn,2 in n2BA22 − 12 , n2BA22 and 0 otherwise. Using

8.8 Proofs

277

the bound for f n (t) we have  f n (2t + 1) − f n (2t) =

2t+1

f n (u)du

2t

 =

2t+1

2t

[ f n (2t) + O(n −2σ −2 )] du.

This and the observation that, for a differentiable function g(x), 

 g(n) = [x2 ]g([x2 ]) − [x1 ]g([x1 ]) −

x1 0. Noting that the functions u i (n) are defined modulo (2Bi ) and that, for a function u(n) ≤ 1 defined modulo q by   u(n)n −2s = (kq)−2s u(n) + O(qk −2σ −1 ) kq≤n 0, as required. Case (ii): Both a and b are odd. When a and b are both odd, any x0 and y0 satisfying ax0 − by0 = 1 are necessarily of opposite parity, so that on the line ax − by = n our parameterization gives (−1)x+y = (−1)n . We here choose our x0 , y0 to satisfy Ai = ri x0 − si y0 > 0 for i = 1, 2 (we can do this by replacing x0 , y0 by x0 + bj, y0 + a j for a suitably small j), and set βi =

Bi . Ai

Hence, altering the order of the n and t summations, we have   A N (T, Q, s) = (−1)n f n (t) n≤N

=



α1 ≤t≤N α1

nα1 ≤t≤nα2

E1 +



N α1 ≤t≤N α2

E2,

280

Madelung sums in higher dimensions

where



E 1 :=

(−1)n f n (t)

tβ2 ≤n≤tβ1



E 2 :=

(−1)n f n (t).

tβ2 ≤n≤N

Just as in case (i) (with the roles of n and t reversed), summing along the line joining tβ1 P1 and tβ2 P2 we have 1 − 2v2 (t) 1 1 − 2v1 (t) 1 Q (β2 P2 )−s − Q (β1 P1 )−s + O(t −2σ −1 ) 2 2 t 2s t 2s −s  B2 1 − 2v2 (t) 1 1 1 − 2w(N ) E2 = Q P2 − f N (t) + O(t −2σ −1 ) 2 A2 2 t 2s t 2s   t B1 t B1 and 0 otherwise, v2 (n) = − 12 , 2A where v1 (n) = 1 if there is an integer in 2A 1 1   t B2 t B2 and 0 otherwise, and w(N ) is 1 if N is 1 if there is an integer in 2A − 12 , 2A 2 2 even and 0 if N is odd. Hence  1 − 2v2 (t)  1 1 A N (T, Q, s) = (−1) N f N (t) + Q (β2 P2 )−s 2 2 t 2s E1 =

N α1 ≤t≤N α2

α1 ≤t≤N α1

 2v1 (t) − 1 1 + Q (β1 P1 )−s + C6 (s) + O(N −2σ ), 2 t 2s α1 ≤t≤N α1

with C6 (s) analytic in Re s > 0. It is not hard to see that the first sum is simply 1 2 A N (l3 ∩ T, Q, s). Hence it remains to verify that the other sums differ from 1 1 2 A N (l2 ∩ T, Q, s) and 2 A N (l1 ∩ T, Q, s) by a function analytic for Re s > 0. One proceeds just as in case (i) (with the roles of Ai and Bi reversed and with 2|Bi if and only if 2 |ri si ), showing that, for i = 1, 2,  1 − 2vi (t) (−1)i Q (βi Pi )−s t 2s αi ≤t≤N αi

 λ(li ) Q (βi Pi )−s t −2s + C7,i (s) + O(N −2σ ) Ai αi ≤t≤N αi  −s = λ(li )Q (Bi Pi ) t −2s + C8,i (s) + O(N −2σ )

=

1/Bi ≤t≤N /Bi

= A N (li ∩ T, Q, s) + C9,i (s) + O(N −2σ ), with C9,i (s) analytic in Re s > 0, and the result is plain. Proof of Theorem 8.7 Part (i) Define d to be the least common multiple of the di and (for a given positive real ν) set N = d[ν/d]. Then ν P and N P differ by at most a finite collection of lines of the form li : ai x − bi y = ci n, ci = 0, with N ≤ n ≤ ν.

8.8 Proofs

281

Hence Aν (P, Q, s) differs from A∗N (P, Q, s) by a finite sum of 12 An (li ∩ P, Q, s) with ci = 0 and N ≤ n ≤ ν, which is exactly 12 A N (li ∩ P, Q, s) for any radial lines (i.e., lines with ci = 0), together with a finite set of points lying at the intersections of these various lines. However, we have already seen (see (8.8.4)) that when ai /di and bi /di are of opposite parity the lines ai x + bi y = nci , ci = 0, contribute O((ci n/di )−2σ ) = O(ν −2σ ) while, for the radial lines, we found (see (8.8.5)) a contribution Ci (s) + O(N −2σ ), with Ci (s) analytic for Re s > 0 and Ci (s) = 0 unless (0, 0) lies on li ∩ P. The left-over points, being of distance  ν from the origin, similarly contribute only terms of size O(ν −2σ ), and so the limit exists for all Re s > 0. Part (ii) Given our polygon with sides li : ai x − bi y = ci we set δ=

1 2

min |ci |−1 i

and observe that, for an integer N , in going from N P to (N + δ)P we may lose some lines of points but can gain no new lattice points and, similarly, in going from (N − δ)P to N P we may gain but cannot lose lattice points. Hence the two differences |A N ±δ (P, Q, δ) − A N (P, Q, δ)| consist solely of sums of A N (li ∩ P, Q, σ ) with ci = 0 (together with odd points of intersection that are of size O(N −2σ )). Further, if we take N = 2nd (where d is the least common multiple of all the gcd(ai , bi )) then every li with ci = 0 will appear in one of these sums (which sum will depend on whether (0, 0) lies on the same or the opposite side of the line as does the interior of the polygon). As in part (i) the lines li with ai /di and bi /di of opposite parity and ci = 0 will only contribute O(N −2σ ). However, if a I /d I and b I /d I are both odd and c I = 0 then A N (l I ∩ T, Q, σ ) = (−1)ci (2nd)/d I



f n (t)

κ1 N ≤t≤κ2 N

will contribute  κ1 N ≤t≤κ2 N

f n (t) ≥ e1



N −2σ ≥ e2 N 1−2α ,

κ1 N ≤t≤κ2 N

for some positive constants ei . Hence, if 0 < σ ≤ 12 and we have at least one l I with a I /d I and b I /d I both odd and c I = 0 then at least one of |A N ±δ (P, Q, δ) − A N (P, Q, δ)| will always be bounded away from 0 by a constant (irrespective of

282

Madelung sums in higher dimensions

N ) and the limit A(P, Q, σ ) cannot exist. When P is a star body centred at (0,0) we set δ = 1 and (since we only gain points in going from (N − 1)P to P) the same argument shows that A N (P, Q, σ ) − A N −1 (P, Q, σ ) does not tend to zero with N ; hence the limit does not exist even if we restrict ourselves (as is natural) just to integer scalings of P. When all the li with ci = 0 have ai /di , bi /di of opposite parity but there are radial lines (l1 through lk say) with ci = 0 and ai /di , bi /di both odd we consider A N (P, Q, σ ) for N = nd. By the theorem it is clear that convergence for 0 < σ ≤ 12 will be determined solely by the sum over the perimeter. As in part (i) the lines li , i > k, cannot disturb the convergence. However, the radial lines li , i = 1, . . . , k, each contribute   A N (li ∩ P, Q, σ ) = f n (t) = f (Pi ) t −2σ κ1 N ≤t≤κ2 N



log N N 1−2σ

≥ e3

t =0

κ1 N ≤t≤κ2 N

if σ = 12 , if σ < 12

(where e3 is some positive constant) and the resulting sum is plainly unbounded as N → ∞. The proof of Corollary 8.3 is immediate from Theorem 8.7. Proof of Theorem 8.8 We split P into a series of cones Pi each of whose base is a face of P and each with vertex (0, 0, 0): Pi := {(x, y, z) : ai x + bi y + ci z ≤ ei , αi j x + βi j y + γi j z ≤ 0, 1 ≤ j ≤ Ji }, for some integers αi j , βi j , γi j , so that A N (P, Q, s) =

I 

A!N (Pi , Q, s),

i=1

A!N (P,

where Q, s) indicates that points on the sides of N P (excepting the base) are to be counted with weight 12 . Slicing up each three-dimensional polygon N Pi into two-dimensional polygons Pi,m parallel to its base, where Pi,m :={(x, y, z) ∈ Z2 : ai x + bi y + ci z =m, αi j x + βi j y + γi j z ≤ 0, 1 ≤ j ≤ Ji }, we have A!N (Pi , Q, s) =

N ei   ∗ m=1 x∈Pi,m di |m

Q( x )−s ,

 where ∗ denotes that points on the boundary of the polygon are to be counted with weight 12 . We suppose now that Pi,m comes from a face with at least one of ai /di , bi /di , or ci /di even. We assume (replacing m by di m as necessary) that gcd(ai , bi , ci ) = 1

8.8 Proofs

283

(reordering as necessary), that ci is odd, and that ai is even. Setting αi = gcd(ai , ci ) and choosing an even integer bi such that bi bi ≡ 1 (mod αi ) and an odd integer (ai /αi ) such that (ai /αi )(ai /αi ) ≡ 1 (mod ci /αi ) we make the change of variables demanded by the relation ai x + bi y ≡ m (mod ci ) on Pi,m , i.e., y = bi m + αi y  and

 x=

ai αi



1 − bi bi αi



 m−

   ai ci bi y  + x . αi αi

Observe that ai x + bi y + ci z = m becomes      1 − (ai /αi ) (ai /αi ) 1 − bi bi ai − x z =m αi ci /αi αi   1 − (ai /αi ) (ai /αi )  y − bi ci /αi and that 



(−1)x+y+z = (−1)x +y . Hence the sum of (x, y, z) over Pi,m is replaced by a sum of (x  , y  ) over m Ri ∩ Z2 , where Ri is the polygon Ri = {(x  , y  ) ∈ Z2 : αi j x  + βi j y  ≤ γij , 1 ≤ j ≤ Ji } and the αi j , βi j , γij are integers with, we shall assume, no common factor; we define di j :=gcd(αi j , βi j ). Writing Q(x, y, z) = Q i (x  , y  , m) (where Q i (x, y, z) is a positive definite quadratic form), we have ∗ ∗ Q(x, y, z)−s = (−1)x+y Q i (x, y, m)−s =: A∗m (Ri , Q i , s), (x,y,z)∈Pi,m

(x,y)∈m Ri ∩Z2

 here ∗ denotes that points on the perimeter are counted with weight 12 . To evaluate the A∗m (Ri , Q i , s) one proceeds almost exactly as in the proof of Proposition 8.1 (replacing the bounds for the kth derivative, k = 0, 1, 2 in (8.8.3) by   | f n(k) (t)| = O (|n| + |m|)−2σ −k and so on), to give 

†

(−1)x+y Q i (x, y, m)−s = O(m −2σ ),

(x,y)∈m Ri ∩Z2

284

Madelung sums in higher dimensions

 where † denotes that for each side li j of the polygon Ri the last line of points parallel to that side,

γij di j Ri j :=  (li j ∩ Ri ) Li j (m) := m Ri j , di j γi j is to be included in the sum with weight 12 . The desired sum A∗ (Ri , Q i , s) thus differs from this latter sum only by the addition or exclusion of the (half-weighted) expressions for those sides li j for which m ≡ 0 (mod di j ); furthermore, A∗m (Ri ,

Q i , s) = O(m

−2σ

Ji 1 )+ Ai j (m), 2 j=1 di j|m

where



Ai j (m) := ±

(−1)x+y Q 1 (x, y, m)−s ,

(x,y)∈Li j (m)

the ± sign being determined by whether (0,0) lies respectively on the interior or exterior side of li j . Thus A!N (Pi , Q, s) = Ci,0 (s) +

Ji 

E i j + O(N 1−2σ ),

j=1

with E i j :=



Ai j (m)

1≤m≤N ei /di di j|m

and Ci,0 (s) analytic for all Re s > 12 . Now, if αij := αi j /di j and βij := βi j /di j are of opposite parity then one readily shows that alternation in sign along the line Li j (m) gives (in the manner of (8.8.4)) Ai j (m) = O(m −2σ ), and hence E i j = Ci j (s) + O(N 1−2σ ) with the Ci j (s) analytic for Re s > 12 . If the αij , βij are both odd, parameterizing the line as in the proof of Proposition 8.1 we have  (−1)x+y Q i (x, y, m)−s An (Ri j , Q i (m), s) := (x,y)∈n Ri j ∩Z2

= (−1)n



nξ1 ≤t≤nξ2

Q 1 (nx0 − αij , βij t − ny0 , m)−s ,

8.8 Proofs

285

for some fixed integers x0 = x0 (i, j), y0 = y0 (i, j), with αij x0 − βij y0 = 1, and rational numbers ξk = ξk (i, j). So, for bounded integers k1 and k2 we certainly have   An+k1 (Ri j , Q i (m + k2 ), s) = (−1)k1 An (Ri j , Q i (m), s) + O (|m| + |n|)−2σ , where trivial bounding gives



An (Ri j , Q 1 (m), s) = O

n (|n| + |m|)2σ

 .

Hence, splitting the sum over m into multiples of 2di j we have   Ai j (2di j l) + O(m −2σ ) + O(N 1−2σ ), E i j = Si j l≤N ei /2di di j

1≤m≤N ei /di

where, as may be readily checked, Si j :=

2di j 



(−1)

mγij /di j



= 0,

m=1 di j |m

giving E i j = Ci j (s) + O(N 1−2σ ) with Ci j (s) analytic for Re s > 12 . Therefore A!N (Pi , Q, s) = Ci (s) + O(N 1−2σ ), with Ci (s) analytic for all Re s > 12 , and part (i) of the theorem follows at once if all the faces have at least one of ai /di , bi /di , ci /di even. For part (ii) one proceeds in the manner of the proof of Theorem 8.2 to show, by counting the number of points on the faces with ai /di , bi /di , ci /di all odd, that (for suitable multiples N and δ := 12 mini ei−1 ) the contribution from those faces to |A N (P, Q, s) − A N −δ (P, Q, s)| does not tend to zero as N → ∞. Proof of Theorem 8.9 Our approach resembles the proof of Theorems 5 and 6 in Borwein et al. [4, Section IV]. We set T (z) := Tm (z) :=

∞  n 1 =−∞ m1  n 1 =−m 1

∞ 

···

q(n 1 , . . . , n k )z Q(n 1 ,...,n k ) ,

n k =−∞ mk 

···

q(n 1 , . . . , n k )z Q(n 1 ,...,n k ) ,

n k =−m k

and define the normalized Mellin transform Ms [ f ] for Re s > 0 by  ∞ Ms [ f ] :=  −1 (s) f (t)t s−1 dt. 0

286

Madelung sums in higher dimensions

We set F(s) := Ms [T (e−t ) − q(0, . . . , 0)] and observe that (since Ms [e−at ] = a −s for a > 0) Am (s) = Ms [Tm (e−t ) − q(0, . . . , 0)]. We shall need the following uniform-boundedness lemma. Lemma 8.3 For any t > 0 and integers Ni ≥ 0 (with at least one Ni > 0)    ∞  ∞     −t Q(n 1 ,...,n k )  −λ(N12 +···+Nk2 )t  S :=  ··· q(n 1 , . . . , n k )e  < Ce n 1 =N1  n k =Nk for some C = C(Q) > 0 and λ = λ(Q) > 0. Proof We set g(u 1 , . . . , u k ) := e−Q(u 1 ,...,u k ) and write χ N for the characteristic function of the region {(u 1 , . . . , u k ) ∈ Rk : u i ≥ Ni }. Applying the partial summation technique employed in the proof of Theorem 8.1, ∞ 

∞   n 

a(n)b(n) =

n=−∞

 a(l) [b(n) − b(n + 1)]

n=−∞ l=−∞ ∞   n 

=−

n=−∞

l=−∞



n+1

a(l)

b (u) du,

n

to each of the variables n i in turn we obtain   ∞ ∞       ··· W (n 1 , . . . , n k )I (n 1 , . . . , n k ) , S=   n 1 =−∞

n k =−∞

where n1 

W (n 1 , . . . , n k ) :=

···

r1 =−∞

nk  rk =−∞

q(r1 , . . . , rk )χ N (r1 , . . . , rk )

and  I (n 1 , . . . , n k ) :=

n 1 +1 n1

 ···

n k +1

nk

√ √ ∂ ∂ ··· g(u 1 t, . . . , u k t) du 1 · · · du k . ∂u 1 ∂u k

8.8 Proofs

287

By assumption, |W (n 1 , . . . , n k )| < B (which vanishes unless n i ≥ Ni ), hence   ∞  ∞ ∂ √ √  ∂  S≤B ··· ··· g(u 1 t, . . . , u k t) du 1 · · · du k  ∂u k N1 Nk ∂u 1   ∞  ∞  ∂  ∂  =B ··· g(u 1 , . . . , u k ) du 1 · · · du k . √ ··· √  ∂u k N1 t Nk t ∂u 1 Now, since Q(x1 , . . . , xk ) is positive definite we have Q(x1 , . . . , xk ) > 2λ(x12 + · · · + xk2 ) for some λ > 0, giving   ∂  ∂ −2λ(u 21 +···+u 2k )    ∂u · · · ∂u g(u 1 , . . . , u k ) ≤ |P(u 1 , . . . , u k )|e 1 k   2 2 = O e−λ(u 1 +···+u k ) , where P(u 1 , . . . , u k ) = PQ (u 1 , . . . , u k ) is some polynomial of total degree k. Thus  ∞  ∞     2 2 −λ(u 21 +···+u 2k ) du 1 · · · du k = O e−λ(N1 +···+Nk )t , S= √ ··· √ O e N1 t

Nk t

as claimed. Observing that (replacing u i by −u i as necessary) T (e−t ) − q(0, . . . , 0) can be written as a sum of sums of the form S with at least one Ni ≥ 1 and that T (e−t ) − Tm (e−λt ) can be written as a sum of sums with at least one Ni ≥ m = min m i , we have   & % 2 T (e−t ) − Tm¯ (e−t ) = O e−λm t . T (e−t ) − q(0, . . . , 0) = O e−λt , Hence

 ∞

F(s) =  −1 (s) T (e−t ) − q(0, . . . , 0) t s−1 dt 0      ∞ (σ )   = O  −1 (s) e−λt t σ −1 dt = O |(s)| 0

exists, and similarly    −1  |F(s) − Am (s)| = O  (s)

0

∞

e

−λm 2 t σ −1

t

 dt

  −2σ (σ ) →0 =O m |(s)|

as m → ∞ in any region {s : σ > ε, |s| ≤ K } for a fixed positive ε and K . Thus the limit exists and is analytic for all Re s > 0.

288

Madelung sums in higher dimensions

Proof of the Section 8.5 formulae We recall the theta functions ∞ 

θ2 (q) :=

q

(n+1/2)2

,

∞ 

θ3 (q) :=

n=−∞

qn

2

n=−∞

and observe (as may be deduced from Zucker and Robertson [24, 25]) that when √ Q( − p) is a unique factorization domain containing u units we have ∞ 

r (n, Q p )q := θ2 (q)θ2 (q ) + θ3 (q)θ3 (q ) = 1 + u n

p

p

n=1

∞ 

(− p|n)

n=1

qn . (1 − q n )

Hence r (n, Q p ), the number of integer representations of n by Q p (x, y), satisfies r (n, Q p ) = u



(− p | d),

d|n

and (8.4.2) is plain. When p = 3 or 7 we can relate r (n, p), the number of integer solutions of √ 2 x + py 2 = n, to r (n, Q p ). Setting N (x + y − p) = x 2 + py 2 and writing Q p (x, y) = 14 (x − y)2 + 14 p(x + y)2 , it is easily seen that r (n, p) represents the number of integer solutions (x, y) of √ N (x + y − p) = n and r (n, Q p ) represents the number of integer solutions √ (X, Y ) of N ((X/2) + (Y/2) − p) = n, with X and Y of the same parity. It is easily seen (matching (x, y) to (X, Y )) that r (4n, p) = r (n, Q p ) and, by congruences modulo 4, that r (n, p) = 0 when n ≡ 2 (mod 4). When n is odd and √ p = 7, congruences modulo 8 show that if N ((X/2) + (Y/2) − p) = n then X and Y are necessarily even (giving a pairing of (x, y) and (X/2, % & and √Y/2)) r (n, 7) = r (n, Q 7 ). When n is odd and p = 3, by putting ω = 1 + −3√ /2 it solutions ((X/2)+(Y/2) −3), is readily checked√that exactly one of the r (n, Q 3 )√ ((X/2) + (Y/2) −3)ω, and ((X/2) + (Y/2) −3)ω2 will be of the form √ (x + y − p), and hence that r (n, 3) = r (n, Q 3 )/3. Thus, setting λ = 1 or 1/3 as p = 7 or 3 respectively, we have ∞  (±1)n r (n, p) n=1

ns

=

∞  r (n, Q p ) n=1

(4n)s

±λ

∞  r (n, Q p ) ns

n=1 n odd

∞

 r (n, Q p ) = 4−s ± λ(1 − 2−s )(1 − (− p|2)2−s ) . ns n=1

8.9 Commentary: Alternating series test

289

So, finally, noting that (−3 | 2) = −1, (−7 | 2) = 1, and (1 − 21−s )ζ (s) = α(s), we have  1 = 2(1 + 21−2s )ζ (s)L −3 (s), 2 + 3y 2 )s (x 2 (x,y)∈Z \(0,0)



(x,y)∈Z2 \(0,0)



(x,y)∈Z2 \(0,0)



(x,y)∈Z2 \(0,0)

(−1)x+y = −2(1 + 21−s )α(s)L −3 (s), (x 2 + 3y 2 )s 1 = 2(1 − 21−s + 21−2s )ζ (s)L −7 (s), (x 2 + 7y 2 )s (−1)x+y = −2α(s)L −7 (s). (x 2 + 7y 2 )s

8.9 Commentary: Alternating series test A mapping a¯ : N N → R is (N -)monotone if, for m 1 , m 2 , . . . , m N ≥ 0, N  

(−1)

 si

a(m ¯ 1 + s1 , m 2 + s2 , . . . , m N + s N ) ≥ 0.

i=1 s1 =0,1

Thus 1-monotonicity corresponds to a(m) ¯ ≥ a(m ¯ + 1), and 2-monotonicity corresponds to a(m, ¯ n) + a(m ¯ + 1, n + 1) ≥ a(m, ¯ n + 1) + a(m ¯ + 1, n), while 3-monotonicity demands that the alternating sum over any unit cube be positive if the bottom corner is positive. We say that a¯ is fully monotone if a¯ and all its restrictions are monotone. Consider N  ∞ 

(−1)

 mi

a(m ¯ 1 , m 2 , . . . , m N ).

i=1 m i =0

(1) As was done in [2], one may prove the following. If a¯ is fully monotone and a( ¯ m) ¯ = 0 then the rectangular sums converge in an alternating fashion. limn→∞ ¯ For instance, in two dimensions one needs to show that sn :=

n 

(−1)i+ j a(i, ¯ j)

i, j=0

satisfies s2n ≥ s2n+2 , s2n+1 ≥ s2n−1 , and s2n − s2n−1 → 0. Drawing a picture helps. Details for two and three dimensions are spelled out in [4, Chapter 2]. (2) Suppose that a¯ is N times continuously differentiable on (R+ ) N . One may show that a¯ is totally monotone if the partial derivatives satisfy a¯ i1 ≤ 0,

a¯ i1 i2 ≥ 0,

a¯ i1 i2 i3 ≤ 0,

...,

(−1) N a¯ i1 i2 ···i N ≥ 0

290

Madelung sums in higher dimensions

for all partial derivatives with i 1 < i 2 < i 3 < . . . < i N .  i+ j+k (i 2 + j 2 + k 2 )− p and Hence, one can check that sums such as ∞ 0 (−1) ∞ i+ j − p (2i + j) converge for p > 0. 0 (−1)

8.10 Commentary: Hurwitz zeta function The notation (k, l : s) introduced in Section 4.2 (see (4.2.5)) can be also expressed in terms of the Hurwitz zeta function ζ (s, q). Here we summarize a few basic properties of ζ (s, q). (1) First note that ζ (s, 1) = ζ (s) and ζ (s, 12 ) = (2s − 1)ζ (s), and we have a multiplication theorem: k ζ (s) = s

k 

ζ (s, m/k).

m=1

By using a Mellin transform, we obtain the following integral representation for ζ (s, q), valid when Re s > 1 and Re q > 0:  ∞ s−1 −qt t e 1 dt, (8.10.1) ζ (s, q) = (s) 0 1 − e−t Equation (8.10.1) can be used to provide an analytic continuation for ζ (s, q) to the whole complex plane except for a simple pole at s = 1. In particular, when s is a non-positive integer we have ζ (−n, q) = −Bn+1 (q)/(n + 1), where Bn (x) denotes the nth Bernoulli polynomial, while when s is a positive integer, ζ (s, q) may be expressed in terms of the polygamma function. Many other special values are known, see for instance [1]; [19] is also a useful reference. When aided by a computer algebra system, (8.10.1) facilitates the rapid evaluation of ζ (s, q), and hence of real and complex Dirichlet L-series, to high precision. In particular, if we make the change of variable x k = e−t in (8.10.1) then  (−1)s−1 1 x l−1 (log x)s−1 d x, (8.10.2) (k, l : s) = (s) 1 − xk 0 which is conducive to both numerical and symbolic computations. For example, suppose that we require to evaluate L i−5 (1) in (4.4.9) at s = 1. Using (8.10.2), we have  1 1 + (1 + i)x + x 2 i d x. L −5 (1) = 2 3 4 0 1+x +x +x +x √ √ &% & % The denominator of the integrand factorizes as x 2 + 1+2 5 x +1 x 2 + 1−2 5 x +1 , so we can split the integrand into partial fractions, and the evaluation follows. For some other L-series the resulting integral may be evaluated using the residue theorem. In all cases, computer algebra systems are a significant aid.

8.10 Commentary: Hurwitz zeta function

291

(2) The n-dimensional Hurwitz zeta function is described in [5]. Let d > 0, a¯ := (a1 , a2 , . . . , a N ), si := sign(ai ), and define a function  ni ∞   si  L a¯ (s, d) := . (8.10.3) ( |ai |n i + d)s {i|ai =0} n i =0

 n −s So, for instance, L −1,2,3 (s, 1) = ∞ n,m, p=0 (−1) (n + 2m + 3 p + 1) . Using a Mellin transform (much as we did for the one-dimensional Hurwitz zeta function), we can show that  1 d−1 x (− log x)s−1 1  d x. (8.10.4) L a¯ (s, d) = ai (s) 0 ai =0 (1 − si x ) N (so If we let A±N (s, d) := ±e¯ (s, d), where e¯ = (1, 1, . . . , 1) ∈ R $ 1 Ld−1 −1 s−1 −N (− log x) (1 ∓ x) d x), then via a Mellin that A±N (s, d) = (s) 0 x transform we have

A±N (s, d) =

∞  (m + N − 1)! (±1)m 1 , (N − 1)! m! (m + d)s

(8.10.5)

m=0

which reduces as a sum of one-dimensional Hurwitz zeta functions after one expands the ratio of factorials. Therefore, using partial fractions on (8.10.4) and also using equation (8.10.5), it transpires that all n-dimensional Hurwitz zeta functions factor into a linear combination of one-dimensional Hurwitz zeta functions. In many cases, we may compute A±N (s, d) recursively. Using integration by parts, we have

∓1 (d − 1)A±(N −1) (s, d − 1) − A±(N −1) (s − 1, d − 1) . N −1 (8.10.6) When d ≥ N , the recursion (8.10.6) reduces A±N to a linear combination of onedimensional Hurwitz zeta functions. When d < N and d ∈ N, the recursion stops at terms of the form A±N (s, 1), which by (8.10.5) evaluates to  N   [N , i] ζ (s − i), A N +1 (s, 1) = N! A±N (s, d) =

i=0

and A−(N +1) (s, 1) has an identical expression except that an extra factor 1 − 21+i−s is inserted into the summand; here [n, k] denotes the (signed) Stirling number of the first kind, which satisfies the generating function n 

[n, k] x k = (x)n = x(x − 1) · · · (x − n + 1)

k=0

and the recurrence [n, k] = (1 − n) [n − 1, k] + [n − 1, k − 1].

292

Madelung sums in higher dimensions

(3) We record some specific evaluations below; some can be computed in more elementary ways than using the full procedure described above. Probably the easiest example is ∞  m,n=1

1 = ζ (s − 1) − ζ (s); (m + n)s

this can be readily seen by the change of variable m + n = k. Some related sums   are n,m (|m| + |n|)−s = 4ζ (s − 1), m,n>0 (−1)m+1 (m + n)−s = ζ (s)/2s , etc. More generally,  n i >0

 (m − 1)! = [m, i] ζ (s − i + 1). s (n 1 + n 2 + · · · + n m ) m

i=1

Other examples include  m,n≥0

 k,m,n>0

(−1)m+n (1 − 2s )(1 − 2s−1 ) L −4 (s) = ζ (s) + , s 2s (2m + n + 1) 2 2 3(1 − 22−s ) (−1)k+m+n+1 1 − 23−s ζ (s − 2) − ζ (s − 1) = s (k + m + n) 2 2 + (1 − 21−s )ζ (s),

and

√  (−1)m+n 2 log 2 1 3π = − − , 3n + m 3 27 6

m,n>0

 (−1)m+n 5π 2 − 27L −3 (2) − 36 log 2 , = 2 108 (3n + m)

m,n>0

where we need to resort to integrals for the last two sums. The integral (8.10.4) also provides identities such as ∞  a1 ,...,an =1

(−1)a1 +···+an +n = (2n , 2n − 1 : s) − 2−ns ζ (s). (a1 + 2a2 + 4a3 + · · · + 2n−1 an )s

References [1] V. Adamchik. On the Hurwitz function for rational arguments. Appl. Math. Comp., 187:3–12, 2007. [2] D. Borwein and J. M. Borwein. A note on alternating series in several dimensions. Amer. Math. Mon., 93:531–539, 1985. [3] D. Borwein, J. M. Borwein, and R. Shail. Analysis of certain lattice sums. J. Math. Anal. Appl., 143:126–137, 1989.

References

293

[4] D. Borwein, J. M. Borwein, and K. Taylor. Convergence of lattice sums and Madelung’s constant. J. Math. Phys., 26:2999–3009, 1985. [5] J. M. Borwein and P. B. Borwein. Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity. Wiley, New York, 1987. [6] J. P. Buhler and R. E. Crandall. On the convergence problem for lattice sums. J. Phys. A: Math. Gen., 23:2523–2528, 1990. [7] M. L. Glasser and I. J. Zucker. Lattice sums. In Theoretical Chemistry, Advances and Perspectives (H. Eyring and D. Henderson, eds.), vol. 5, pp. 67–139, 1980. [8] G. H. Hardy and M. Riesz. The General Theory of Dirichlet Series. Cambridge Tracts in Mathematics and Mathematical Physics, Cambridge University Press, Cambridge, 1915. [9] M. N. Huxley. Exponential sums and lattice points II. Proc. London Math. Soc., 66:279–301, 1993. [10] E. Krätzel. Bemerkungen zu einem Gitterpunktsproblem. Math. Ann., 179:90–96, 1969. [11] E. Krätzel and W. Nowak. Lattice points in large convex bodies, II. Acta Arith., 62:285–295, 1992. [12] E. Landau. Zur analytischen Zahlentheorie der definiten quadratischen Formen (über Gitterpunkte in mehrdimensionalen Ellipsoiden). S. B. Preuss. Akad. Wiss., 458–476, 1915. [13] E. Landau. Über Gitterpunkte in mehrdimensionalen Ellipsoiden. Math. Zeit., 21:126–132, 1924. [14] E. Landau. Über Gitterpunkte in mehrdimensionalen Ellipsoiden. Zweite Abhandlung. Math. Zeit., 24:299–310, 1926. [15] E. Landau. Vorselungen über Zahlentheorie, vol. 2., Part 8, Chapter 6. Chelsea, New York, 1955. [16] B. Novák. Über eine Methode der -abschätzungen. Czech. Math. J., 21:257–279, 1971. [17] B. Novák. New proofs of a theorem of Edmund Landau. Acta Arith., 31:101–105, 1976. [18] M. Riesz. Sur un théorème de la moyenne et ses applications. Acta Univ. Hungaricae Franc.-Jos., 1:114–126, 1923. [19] H. M. Srivastava and J. Choi. Series Associated with the Zeta and Related Functions. Kluwer Academic, Dordrecht, 2001. [20] A. Walfisz. Über Gitterpunkte in mehrdimensionalen Ellipsoiden. Math. Zeit., 19:300–307, 1924. [21] A. Walfisz. Convergence abscissae of certain Dirichlet series. Akad. Nauk Gruzin. SSR. Trudy Tbiliss. Mat. Inst. Razmadze, 22:33–75, 1956. [22] G. N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, 1922. [23] J. R. Wilton. A series of Bessel functions connected with the theory of lattice points. Proc. London Math. Soc., 29:168–188, 1928. [24] I. J. Zucker and M. M. Robertson. Some properties of Dirichlet L-series. J. Phys. A, 9:1207–1214, 1976. [25] I. J. Zucker and M. M. Robertson. Further aspects of the evaluation of  2 2 −s (m,n=0,0) (am + bmn + cn ) . Math. Proc. Camb. Phil. Soc., 95:5–13, 1984.

9 Seventy years of the Watson integrals

Watson [59] published in 1939 the evaluation of three integrals submitted to him, which had arisen from a problem in physics [57]. Over the years these integrals have continued to occur in other aspects of physics such as random walk problems. This article reviews these integrals and their generalizations over the past 70 years.

9.1 Introduction Here we give a short history of three triple integrals and their generalizations; the integrals first appeared some 70 years ago. This provides an example of how physicists and mathematicians like to elaborate particular results and extend them to more general areas. In doing so new techniques are developed, which eventually become standard methods. It is also an example of how chance plays a part in bringing together researchers with knowledge in different fields who collaborate to produce results which might not otherwise have been obtained. This will be a rather personal account, as many hundreds of papers and articles concerning these integrals have been published, and it is not possible in limited space to refer to every contribution, so at the start we apologize to authors whose work we do not cite. The three original integrals are  π π π d x d y dz 1 , WB = 3 π 0 0 0 1 − cos x cos y cos z  π π π d x d y dz 1 WF = 3 , π 0 0 0 3 − cos x cos y − cos y cos z − cos z cos x  π π π d x d y dz 1 WS = 3 . π 0 0 0 3 − cos x − cos y − cos z

(9.1.1) (9.1.2) (9.1.3)

9.1 Introduction

295

These integrals made their first appearance in a paper on magnetic anisotropy by van Pepye [57] in 1938. They emerged in connection with three well-known cubic structures formed by real crystals, namely the body-centred (B), face-centred (F), and simple cubic (S) lattices. Van Pepye was a student of the Dutch physicist H. A. Kramers, and clearly these integrals intrigued the latter. He sent them on to R. H. Fowler, the son-in-law of Rutherford, in Cambridge. Let Watson [59], who worked at Birmingham University, describe in his own words how the problem reached him: The problem of evaluating them was proposed by Kramers to R. H. Fowler who communicated them to G. H. Hardy. The problem then became common knowledge first in Cambridge and subsequently in Oxford, whence it made the journey to Birmingham without difficulty.

Whatever his motive for this last somewhat barbed comment, Watson found closed forms for these integrals and they are now universally known by his name. They will be referred to as WIs. The first, as Watson acknowledged, is fairly well known. Indeed, van Pepye himself had evaluated it and the result can be traced back to Kummer [40]; also, its generalization is simple to find (Maradudin et al. [42]), so the closed form for this more general version of (9.1.1) can now be given. We have  π π π d x d y dz 1 W B (wb ) = 3 π 0 0 0 wb − cos x cos y cos z (  ' 1 1 4 2 (9.1.4) − 1 − wb−2 . = 2K 2 2 π When wb = 1, W B (1) =

1 4 1 4 2 1  K √ =  4 = 1.3932039297 . . . , 2 2 π 4π 3

(9.1.5)

where K is the complete elliptic integral of the first kind. As complete elliptic integrals of the first kind play a primary role in all aspects of WI and their generalizations, it is apposite here to give a short account of them. The complete elliptic integral of the first kind is defined by  1  π/2 dt dx = (9.1.6) K (k) :=

1/2 2 (1 − k 2 sin t)1/2 0 0 (1 − x 2 )(1 − k 2 x 2 ) for 0 ≤ k < 1, where k is known as the modulus. The complementary modulus k  is defined by the relation k 2 + k 2 = 1 and we write K (k  ) := K  (k). The integrals K and i K  play roles in the theory of the Jacobian elliptic functions that are equivalent to that played by π/2 in the theory of the circular functions;

296

Seventy years of the Watson integrals

namely, they are quarter periods. An important property of K (k) is that it can be expressed as a hypergeometric function:

 ∞  1 1 (2n)! 2 2n , π π 2 2 2 k , (9.1.7) K (k) = 2 F1 ;k = 2 1 2 22n (n!)2 n=0

which will later be seen as crucial in our further analysis. Although (9.1.1) and its generalization were easily solved, (9.1.2) and (9.1.3) are a different matter. Watson’s approach [59] was to use an inspired sequence of changes of variable and integrations to reduce the triple integrals to single integrals involving K . Then, using expansions of K and K  that he himself had obtained 30 years previously [58] he was able to evaluate these last integrals. One should go to the original paper to admire the ingenuity displayed in finding (9.1.8) and to enjoy the brilliance of his derivation of (9.1.9). His results are   √  √ 6 1 3 3 3 3−1 (9.1.8) = 14/3 4 = 0.4482203944 . . . , WF = 2 K 2 √ π 2 π 2 2 √ √ √ √  √ √ 4(18 + 12 2 − 10 3 − 7 6) 2  (2 − WS = K 3)( 3 − 2) π2 = 0.5054620197 . . . . (9.1.9) It will be observed that whereas W B and W F are expressed in terms of gamma functions, this was not done for W S and had to await a later investigation. The next appearance of WIs in the literature (though not by name) followed almost immediately. McCrea and Whipple, in an odd paper [44], took up the problem of the ‘drunkard’s walk’ in three dimensions. The strangeness of the paper was not in its content but in the complete obliviousness of the authors to any other work in this field. There were only two references given. The first was to a note of McCrea in the Mathematical Gazette [43], in which he had considered the two-dimensional square lattice and, as a kind of afterthought, the second was to a paper by Courant et al. [13], which had been brought to their attention. The problem may be formulated as follows. An intoxicated person drops his house keys under a lamppost – the origin. Suppose the lamppost is one of an infinite number of posts equally spaced along a line. The inebriate staggers from lamppost to lamppost with an equal probability of going to the right or left while looking for the keys. What is the probability that he or she will return to the origin? The same question may be asked if the lampposts form a two-dimensional square array or a three-dimensional cubic array. In fact this problem was first discussed by Pòlya [48]. He showed that in the one- and two-dimensional cases the probability of return to the origin was unity, i.e., the keys would eventually be found. However, in three dimensions the probability was less than one, but Pòlya did not find a value for it.

9.1 Introduction

297

This problem is mirrored in one dimension exactly by the tossing of a coin (heads or tails), in two dimensions by the throwing of a tetrahedral die, and in three dimensions by the casting of a common six-sided die. Stewart [55] discusses this in some detail. He points out that in tossing a fair coin the a priori probability is equal numbers of heads and tails. As one continues tossing, an imbalance between the numbers of heads and tails will occur but, no matter how large this imbalance becomes, if you carry on flipping the coin the imbalance will eventually correct itself. Similarly, in throwing an unbiased tetrahedral die, the digits 1, 2, 3, and 4 have an equal probability, of 1/4 of appearing. As before one starts with equal numbers of the four digits and again, as one continues throwing, deviations from the initial state will appear. But, as in the coin case, if one continues to throw the four-sided die, the state of equal numbers for the four digits will eventually reoccur. However, in throwing an ordinary six-sided die, each of the digits 1–6 has a probability of 1/6 of occurring. Starting from equal numbers of the six digits, imbalances will develop, but the probability of returning to equal numbers of the six digits is less than unity. Stewart actually credits Ulam as proving this, but gives no reference, and we think that Pòlya [48] must be credited as being first with this result. However, it was McCrea and Whipple who first set out to find a value for this probability. Clearly they did not know of Pòlya’s paper, for no reference to him is made, but they succeeded in showing that the probability of return to the origin was 1 − 1/(3W S )! (We have converted their result into the notation used here. All authors in this field use their own symbols, so all formulae given here are translations of their results into notation consistent with that used here.) Just as McCrea and Whipple were unaware of Pòlya they also were in ignorance of Watson, and it is interesting to follow their attempt to evaluate 3W S . They rapidly reduced W S to a single integral from π/6 to π/2 of a complete elliptic integral multiplied by an angular factor and then resorted to numerical integration. Such was the inaccuracy of their computation that they only gave the value to two significant figures, 3W S = 1.53, which, compared with the value 1.51638606 found from (9.1.9), is woefully in error. They thus found the probability of returning to the origin was 0.34 – surprisingly small. This seems to be the very first numerical evaluation of this probability. A much more accurate result was found by Domb [18] in a completely different manner with no reference to W S whatsoever. He calculated directly the probability of the return to the origin, pn , after n steps. For n odd you cannot return to your starting point and thus p2n+1 = 0. For small n, p2n can be calculated exactly from a rather complex formula. In this way he directly calculated p2 , . . . , p18 ; then, applying Euler–Maclaurin summation to the asymptotic formula for p2n , he obtained an excellent estimate 1.51639, for the equivalent of 3W S , giving 0.34054 for the probability of return to the origin. If we use the more accurate value of Watson, the probability of return to the origin is 0.340537330.

298

Seventy years of the Watson integrals

Now, the Courant paper referred to by McCrea and Whipple was actually a seminal paper on how limiting forms of difference equations could equal the solution of the equivalent differential equation. It was noted that the random walk problem was set up as a difference equation and the solution was none other than the Green’s function of the equation u = 0. This idea was elaborated on by Duffin [20], who considered an infinite lattice in which every point was connected to its nearest neighbour by a unit resistance. If a current is introduced at some lattice point then it will split equally among all connections, and so on at every lattice point, and this again mimics the random walk problem. Duffin then considered an infinite simple cubic lattice in which every lattice point is labelled (l, m, n); each of l, m, and n may take on every integer value from −∞ to ∞. A unit current is introduced into the source point (0, 0, 0). The Green’s function G(l, m, n) is defined by the solution to DG(0, 0, 0) = −1,

DG(l, m, n) = 0 otherwise,

where D is the difference operator corresponding to the Laplacian differential operator  in three dimensions. The solution is found to be G(l, m, n) =

1 π3



π



0

π



0

π

0

cos lx cos my cos nz d x d y dz. 3 − cos x − cos y − cos z

(9.1.10)

This result was ascribed to Courant [12]. A similar result was then obtained by Davies [14] without reference to Duffin or Courant. Davies noted that the resistance between the source point and a point infinitely far away was just G(0, 0, 0); this is of course just the Watson integral (9.1.3). Also, the resistance between the source point and any other point (l, m, n) on the lattice was Rl,m,n = G(0, 0, 0) − G(l, m, n). An excellent modern account of the relationship between random walks on lattices and resistance networks has been given by Doyle and Snell [19]. The association of WIs as special values of Green’s functions of a particular lattice has led to the term Green’s function being applied to simple generalizations of WIs. Thus it has became customary to refer to any integral of the form 1 π3



π 0



π 0



π 0

d x d y dz , w − φ(cos x, cos y, cos z)

(9.1.11)

where φ(cos x, cos y, cos z) is some ternary trigonometric polynomial, as a lattice Green’s function. However, it would be better to think about objects having the form (9.1.11) as functions of a complex variable w. What Watson did was to evaluate the original integrals at a certain critical value of w relevant to each integral, where the integrand becomes infinite at the origin. It thus became a challenge to extend his analysis to general w, i.e., to evaluate

9.2 Solutions for W F (w f ), W F (α f , w f ), and W S (ws ) 1 W F (w f ) = 3 π W S (ws ) =

1 π3



π 0



π

0 π

0



 0



π

0 π

 0

π

299

d x d y dz , w f − cos x cos y − cos y cos z − cos z cos x (9.1.12) d x d y dz . (9.1.13) ws − cos x − cos y − cos z

Then (9.1.12) defines a single-valued analytic function in the w f plane, provided that a cut is made along the real axis −1 < w f ≤ 3. A similar property holds for (9.1.13) provided that the cut on the real axis is made for −3 < ws ≤ 3. Also, rather than having a different w for each lattice, many authors considered a modified form of WI, namely  π π π d x d y dz 1 , (9.1.14) P(u) = 3 π 0 0 0 1 − (u/d)φ(cos x, cos y, cos z) where d is the number of terms in φ. Sometimes it is more convenient to deal with P. It is the evaluation of such quantities that has exercised investigators from the mid 1950s to present times. The first extension of Watson’s result was achieved by Montroll [47]. He considered the integral  π π π d x d y dz 1 W S (2 + α, α) = 3 π 0 0 0 2 + α − α cos x − cos y − cos z and, following exactly the same path as Watson, Montroll found √ 25/2 k1 k2 K (k1 )K (k2 ) W S (2 + α, α) = , (9.1.15) √ απ 2 where  √ √ 1 √ ( 4 + 2α − 2)(2 1 + α − 4 + 2α) , k1 = 2α  √ √ 1 √ k2 = ( 4 + 2α + 2)(2 1 + α − 4 + 2α) . 2α Although this indicated that further generalizations might be possible, no new methods were introduced and the chances of improving on Watson’s ingenuity seemed small. Two big advances were made, however, when Iwata [29] produced a closed-form solution for W F (w f ) and when Joyce [31, 32] further generalized W F but more importantly found a closed form for W S (ws ).

9.2 Solutions for W F (w f ), W F (α f , w f ), and W S (ws ) Iwata’s method of finding a closed form for (9.1.12) was as follows. He copied Watson in reducing the triple integral to a single integral, but did it in an entirely different fashion. He made use of two successive well-known integrations, and then solved this last integral in such a way that this has become a

300

Seventy years of the Watson integrals

standard integration. The first integration was accomplished using the established result  π π dz = 2 , (9.2.1) a − b cos z (a − b2 )1/2 0 which after integrating (9.1.12) over z gives W F (w f ) =

1 π2



π 0



π 0

dx dy . [(w f − cos x cos y + cos x + cos y)(w f − cos x cos y − cos x − cos y)]1/2

Then substituting cos y = u, after some re-arrangement one obtains  π dx W F (w f ) = 2 x − 1)1/2 (cos 0   −1/2  1  w f + cos x w f − cos x 2 −u − u (1 − u ) × du. cos x − 1 cos x + 1 −1 (9.2.2) The standard result  1 −1

du 2 = K (k), 2 1/2 [(a − u)(b − u)(1 − u )] [(a − 1)(b + 1)]1/2

(9.2.3)

where k2 =

2(a − b) , (a − 1)(b + 1)

is now employed on (9.2.2) and after some algebra W F (w f ) is obtained as a single integral, namely ⎡ ' ⎤  π 2 w f + cos2 x 2 ⎦ d x. W F (w f ) = 2 K⎣ (9.2.4) wf + 1 π (w f + 1) 0 There are a number of other ways in which this integral may be expressed; thus for example  π   √ 2 K A + B cos x d x, (9.2.5) W F (w f ) = 2 π (w f + 1) 0 4w f + 2 2 A= , B= , (w f + 1)2 (w f + 1)2  π   2 W F (w f ) = 2 K a 2 cos2 x + b2 sin2 x d x, (9.2.6) π (w f + 1) 0  √ 2 wf 2 wf + 1 a= , b= . wf + 1 wf + 1

9.2 Solutions for W F (w f ), W F (α f , w f ), and W S (ws )

301

Iwata chose to work with (9.2.6), using the representation of K as a hypergeometric function; thus  π  K ( a 2 cos2 x + b2 sin2 x) d x 0

=

  ∞  (2n)! 2 π 2 π (a cos2 x + b2 sin2 x)n d x. 2 22n (n!)2 0

(9.2.7)

n=0

The integral on the right-hand side of (9.2.7) can be evaluated as an infinite sum, and the double sum so produced was identified as F4 , an Appell hypergeometric function of two variables; see Whittaker and Watson [60], p. 300. In this case a theorem derived by Bailey [3] showed that this F4 is in fact the product of two complete elliptic integrals. It is more straightforward to express the result in terms of (9.2.5):  π √ K ( A + B cos x) d x = 2K (k+ )K (k− ), 0

2 2k± =1±



A2 − B 2 −

 (1 − A)2 − B 2 .

(9.2.8)

This result may now be accepted as a standard form. Hence, for W F (w f ) we have W F (w f ) = 2k 2f ±

=1±

4 π 2 (w

√ 4 wf

f

K (k f + )K (k f − ),  (w f − 1)( w f − 3) − . (w f + 1)3/2

+ 1)

(w f + 1)3/2

(9.2.9)

A further advance was made by Joyce [31], who published a generalization of Iwata’s work. Thus he solved  π π π d x d y dz 1 , W F (α f , w f ) = 3 π 0 0 0 w f − α f cos x cos y − cos y cos z − cos z cos x (9.2.10) the solution being W F (α f , w f ) = where

4 K (k+ )K (k− ), π 2 (w f + α f )

 1/2 4α f w f 1 1 + wfαf (w f + α f )2

1/2

1/2 wf − αf w f + (2 − α f ) w f − (2 + α f − . 2 (w f + α f ) (9.2.11)

2 2k± (α f , w f ) = 1 ±

His analysis involved showing that W F (α f , w f ) could be expressed as  ∞ J0 (at)J0 (bt)J0 (t) dt, (9.2.12) 0

302

Seventy years of the Watson integrals

where a and b are functions of w f and α f and J0 is a Bessel function. A theorem of Bailey [4] then enabled (9.2.12) to be evaluated directly. Joyce indicated that he would describe the process in detail later but he never did. Instead, 32 years later, under the prompting of colleagues the result was republished but was shown to be derivable by Iwata’s method. We are assured that the promised other proof will be forthcoming. Rather interestingly, in his evaluation of W S , Watson [59] had employed the same strategy as Iwata in evaluating his last integral using the method of Appell and Bailey, but his result was confined to a special case. Also, Iwata ends his paper with the remark that W S might have a similar result to (9.2.6). Now one can be assured that Iwata must have tried his technique on W S (ws ), since the latter has a considerably simpler form than W F (w f ), but if indeed one performs the first two integrations one obtains  π π π d x d y dz 1 W S (ws ) = 3 π 0 0 0 ws − cos x − cos y − cos z    π 2 2 1 K d x. (9.2.13) = 2 ws − cos x π 0 ws − cos x This form was actually obtained by Tikson [56]. However, this does not yield to Iwata’s approach and it seems that a different technique is required to obtain a closed form for W S (ws ). This was accomplished by Joyce [32] in a classic paper, and a brief description of his method is now given. The procedure used by Joyce was entirely different to that of Iwata. He began by considering the P form of the simple cubic,      π π π 3 d x d y dz 1 3 WS . = PS (u) = 3 u u π 0 0 0 1 − 13 u (cos x + cos y + cos z) (9.2.14) By expanding the integrand in (9.2.14) in powers of u, a power series for P(u) was found. Thus ∞  pn u n , |u| < 1, (9.2.15) PS (u) = n=0

where pn =

1 π3

 0

π

 0

π



π 0



1 3 (cos x

+ cos y + cos z)

2n

d x d y dz.

(9.2.16)

In the theory of random walks, the coefficient pn gives the probability that a random walker will return to his starting point (not necessarily for the first time) after a walk of 2n steps on a SC lattice. For n → ∞ the behaviour of pn is described by an asymptotic formula given by Domb [18], which showed that the range of

9.3 The Watson integrals between 1970 and 2000

303

validity of (9.2.15) could be extended to |u| = 1. So, PS (1), which is equal to 3W S , may be written as follows: PS (1) =

∞ 

pn .

(9.2.17)

n=0

Joyce then proceeded to find a closed form for pn and to establish a three-term recurrence relation amongst the pn . This enabled him to construct the following third-order differential equation for PS : d 3 PS d 2 PS 2 + 12x(2x − 15x + 9) dx3 dx2 d PS + 3(9x 2 − 44x + 12) + 3(x − 2)PS = 0, dx

4x 2 (x − 1)(x − 9)

(9.2.18)

where x = u 2 . A remarkable result of Appell [1] allowed this third-order equation to be reduced to a second-order equation, and the solutions of this were identified as Heun functions. Then applying various standard transformation to these, they were turned into hypergeometric functions and finally produced the important result √ 4 − 3t 2 K (k+ )K (k− ), W S (ws ) = 2 π ws 1 − t √ √ 2 2k± = 1 ± 12 v 4 − v − 12 (2 − v) 1 − v, (9.2.19) ' 2ws2 t = ws2 + 3 − (ws2 − 9)(ws2 − 1), v = t/(t − 1). We have set out here in a few words an investigation which occupies almost 40 printed pages, and once again we suggest that the original paper be viewed in order to appreciate the effort put into obtaining (9.2.19).

9.3 The Watson integrals between 1970 and 2000 In the years that followed Iwata’s solution for W F (w f ), several authors, e.g., Glasser [21], Hioe [28], Rashid [50], and Montaldi [46], investigated increasingly complex forms of (9.1.11). All yielded to Iwata’s method. As an example we give Glasser’s [21] form of (9.1.11): φ = cos x cos y cos z+cos x cos y+cos y cos z+cos z cos x +cos x +cos y+cos z, (9.3.1) with solution  π π π 4 d x d y dz 1 = 2 K 2 (k), 3 π 0 0 0 w−φ π (w + 1)   1 w − 7 1/2 2 where k = . (9.3.2) 1− 2 w+1

304

Seventy years of the Watson integrals

What real lattice this represents is open to question, but it is a tribute to Iwata’s approach that such a complex integral was evaluated by his method. At around this time, Zucker stumbled accidentally into WIs and this led to his first contribution. To explain how this occurred, more information on complete elliptic integrals needs to be given. It had been shown by Abel that when √ K  (k) a+b N (9.3.3) = √ , K (k) c+d N with a, b, c, d, N all integers, then k is a root of an algebraic equation with integral coefficients which can be solved in radicals. In the particular case where √ K  /K = N the k(N ) are called singular values, and for convenience K [k(N )] will be denoted by K [N ]. It has long been known for N = 1, 3, and 4 that K [N ] can be expressed in terms of gamma functions of rational arguments, the values for K [1] and K [3] having been found by Legendre (see [60], √ p. 524). Now, the result given in (9.1.5) for W B contains the square of K (1/ 2), which is  indeed equal to K [1], and the result (9.1.6) for W F contains the square of √ √ K ( 3 − 1)/2 2 which is K [3]. Of course Watson knew this; hence his translation of the results in terms of K into gamma functions. However, the K involved in W S is actually K [6] and its expression in gamma functions √ was unknown. Glasser and Wood [23] had actually given K [2], for which k = 2 − 1, in terms of gamma functions. This might have been extracted from a result of Ramanujan [50],√ who had actually found the complete elliptic integral of the second kind for k = 2 − 1 but no one seemed to have noticed. Now, in investigating properties of the double sums ∞ 

(am 2 + bmn + cn 2 )−s ,

(m,n=0,0)

a comparatively straightforward procedure gave K [N ] in terms of gamma functions for many N ; see Zucker [61]. Indeed, K [N ] for N = 1, . . . , 16 (excluding N = 14, which did not succumb to this technique) were thus evaluated. A paper by Selberg and Chowla [54] proved that all K [N ] could – with sufficient labour – be evaluated in terms of gamma functions, but they only gave explicit values for N = 5 and N = 7. Thus the values found were new and included N = 6. Together, Glasser and Zucker found [24] √ 4 6  1   5  % 7 &  11  (9.3.4) W S = 2  24  24  24  24 , π which unfortunately was wrong – a factor of 384π had been omitted. While it is almost certain that everyone at some time has produced a wrong sign in a calculation or has lost a factor 2 somewhere, 384π requires an explanation. The reason is depressingly simple. The result found for the square of K [6] was

9.3 The Watson integrals between 1970 and 2000

K 2 [6] =

305

    % &   √ √ √ √ 1 5 7  24  24 ( 2 − 1)( 3 + 2)(2 + 3) 24  11 24 384π

(9.3.5)

√ √ and, was substituted + 12 2 − 10 3 − √ this √ √ √into (9.1.9), the fact that (18√ √ when 7 6)( 2−1)( 3+ 2)(2+ 3) dramatically collapsed to 6 made the authors forget the 384π in the denominator of (9.3.5) in the desire to publish quickly. So, the correct result is √ 6  1   5  % 7 &  11  (9.3.6)  24  24  24  24 , WS = 96π 3 as is pointed out by everyone who quotes the original. Later, Borwein and Zucker [6], making use of the surprising relation     1  11  24 √ √ 24   % & = 3(2 + 3)1/2 , 5 7  24  24 allowed the reduction of (9.3.4) to the even more compact form √ ( 3 − 1) 2 1 WS = [( 24 )( 11 24 )] . 96π 3

(9.3.7)

A more substantial improvement to the theory was made by Joyce [33, 34] – which considerably simplified Iwata’s result (9.2.9) for W F (w f ) and his own expression (9.2.19) for W S (ws ). In both cases the original results appeared as the products of two complete elliptic integrals with different moduli. In the abovereferenced papers it was shown that the results could be expressed as the square of a single K . In [34], by using the elliptic modular transformation of order 3, the solutions were found in a particularly simple fashion. They are given below in parametric form: 2  4ξ(1 − 3ξ )(1 + ξ ) 2 K (k) , (9.3.8) W F (w f ) = (1 − ξ )3 (1 + 3ξ ) π where k2 = and

16ξ 3 , (1 − ξ )3 (1 + 3ξ )

 ξ=

wf + 1 − wf − 3 +

√ √

wf wf

,

2  2 (1 − 9ξ 4 ) K (k) W S (ws ) = ws (1 − ξ )3 (1 + 3ξ ) π

with 16ξ 3 , k2 = (1 − ξ )3 (1 + 3ξ )

8  9 9 ws − w 2 − 1 :  s ξ= . ws + ws2 − 9

(9.3.9)

306

Seventy years of the Watson integrals

Now define the sets of points in the w f and ws cut planes by C f and Cs respectively. There is insufficient space to expand upon the regions of C f and Cs over which Iwata’s and Joyce’s original results are valid. Suffice it to say that their range was severely limited in those sectors. The new results were valid for all C f and Cs . The complicated structures of (9.3.8) and (9.3.9) may be simplified by applying various 2 F1 transformation formulae to the complete elliptic integrals in these formulae. For example, Delves and Joyce [15] showed that

2   1 3 , 2ws − ws2 − 9 8 8 ; t (w ) , (9.3.10) W S (ws ) = 2 F1 s 1 ws2 + 3 where t (ws ) =

 2 ' 16 2 2 2−9 w (w − 5) − (w − 1) w . s s s s (ws2 + 3)4

This result may also be used to calculate W S (ws ) at any point in Cs . For ws = 3 the original Watson integral becomes 

2 1 3 ,8 1 1 8 ; . (9.3.11) W S = 2 2 F1 1 9 Then making use of a famous result of Clausen [10], this may be written as 

 1 1 3 , 2, 4 1 1 4 ; W S = 2 3 F2 . (9.3.12) 1, 1 9 From this and (9.3.7), we find the striking result that √ ∞  (4n)! 1 3 − 1   1   11 2  24  24 = . (n!)4 (48)2n−1 π3

(9.3.13)

n=0

9.4 The singly anisotropic simple cubic lattice The equation W S (αs , ws ) =

1 π3

 0

π

 0

π



π 0

d x d y dz , ws − αs cos x − cos y − cos z

(9.4.1)

is a further generalization of the original Watson integral W S . In terms of random walks on a cubic lattice, the spacing between lattice points in one of the dimensions is different from that of the other two. It might be better to refer to (9.4.1) as the Watson integral for a tetragonal lattice. The investigation of Delves and Joyce [15] in finding a closed form for (9.4.1) is an exceptional work of analysis, which again can be described only briefly here. The integrand in (9.4.1) was expanded in powers of 1/ws and the resulting series integrated term by term. Thus

9.4 The singly anisotropic simple cubic lattice ws W S (αs , ws ) = y =

∞ 

μ2n (αs )z n ,

307 (9.4.2)

n=0

where z = 1/ws2 , and  π π π 1 μ2n = 3 (αs cos x + cos y + cos z)2n d x d y dz. π 0 0 0

(9.4.3)

An explicit expression was found for μ2n (αs ) and a complex recurrence relation established amongst μ2n+2 (αs ), μ2n (αs ), μ2n−2 (αs ), μ2n−4 (αs ), and μ2n−6 (αs ). From this it was shown that y is the solution of a sixth-order differential equation L 6 (y) = 0, where L 6 is an extremely long and intricate operator. These authors then showed it is possible to express L 6 as the product of a fourthorder differential operator, L 4 , and a second order differential operator and thus to reveal that y is the solution of L 4 only. A further step then allowed the solutions of L 4 (y) = 0 to be expressed as a product of the solutions of two second-order differential equations. Then Schwarzian transformation theory enabled both these second-order differential equations to be reduced to the standard Gauss hypergeometric differential equation. Finally, the hypergeometric functions which are solutions of these equations could be transformed by well-known quadratic transformations into complete elliptic integrals as in (9.1.9). The final solution for y is:  ws W S (αs , ws ) = 2[1 − (2 − αs )2 z + 1 − (2 + αs )2 z]−1/2    2 2 K (k+ ) K (k− ) , × (9.4.4) π π where 2 2 2k± ≡ 2k± (αs , z) −3   1 − (2 − αs )2 z + 1 − (2 + αs )2 z =1− ' ' √ √ × 1 + (2 − αs ) z 1 − (2 + αs ) z  ' ' √ √ + 1 − (2 − αs ) z 1 + (2 + αs ) z  ' ' ' √ √ × ±16z + 1 − αs2 z 1 + (2 − αs ) z 1 − (2 + αs ) z

+

'

 ' √ 1 − (2 − αs ) z 1 + (2 + αs ) z . √

(9.4.5) This is a few-word summary of some 60 pages of analysis and, for the third time here, we recommend readers to view the original paper to appreciate the tenacity that went into producing (9.4.4) and (9.4.5), to which the comment of Bornemann et al. [5] may be added: ‘What a triumph of dedicated men; for such problems current computer algebra systems are of little help’.

308

Seventy years of the Watson integrals

Is there a more direct approach to (9.4.5)? It turns out that there is, but it is doubtful whether it would have been found without the inspiration of the known result. Going back to Tikson’s result (9.2.12) and putting in the anisotropy does not complicate the last integral very much. In fact, doing the first two integrals using Iwata’s approach, one obtains    π 2 2 1 K d x. (9.4.6) W S (αs , ws ) = 2 ws − αs cos x π 0 ws − αs cos x Now the following substitution is made, cos x =

ws cos ψ + αs , ws + αs cos ψ

and a much simplified version of (9.4.6) appears, namely  π 2 1 W S (αs , ws ) =  K (C + D cos ψ) dψ, ws2 − αs2 π 2 0

(9.4.7)

(9.4.8)

where C=

2ws − αs2

ws2

and

D=

2αs . ws2 − αs2

(9.4.9)

In going through the first two steps of the Iwata process for the anisotropic facecentred cubic lattice, we arrived at  π √ 2 1 W F (α f , w f ) = K ( A + B cos x) d x, (9.4.10) 2 wf + αf π 0 where A=

2(2α f w f + 1) (w f + α f )2

and

B=

2 , (w f + α f )2

(9.4.11)

and the similarity between (9.4.8) and (9.4.10) is evident. We have of course the solution of (9.4.10) in (9.2.11), and it seemed natural to look for a solution to (9.4.8) along the same lines as Iwata had done for (9.4.10). This was accomplished as follows. For convenience we write  π √ 2 K ( A + B cos x) d x, I (A, B) = 2 π 0  π 2 J (C, D) = 2 K (C + D cos ψ) dψ. (9.4.12) π 0 As Iwata did, the integrand of I (A, B) is expanded and integrated term by term to yield a double series. However, instead of identifying this double series as an Appell hypergeometric function, I (A, B) is represented by a Kampé de Feriet series. Doing the same thing to J (C, D) yields a different Kampé de Feriet series but, by applying appropriate transformation formulae to the I (A, B) series, it can

9.4 The singly anisotropic simple cubic lattice

309

be put into the form for J (C, D). The following remarkable connection formula was found: J (C, D) = (1 − A + B)1/2 I (A, B),

(9.4.13)

where A and B are appropriate solutions of the simultaneous equations C2 = −

(A − B)(1 − A + B) , (1 − A + B)2

D2 =

2B . (1 − A + B)2

(9.4.14)

By finding relevant solutions for A and B and substituting into (9.4.13), after considerable algebraic simplification a standard form for J (C, D) may now be established. It is    2 2 −1/2 2 2 2 K (k+ )K (k− ), J (C, D) = 2[(1 − D) − C + (1 + D) − C ] π (9.4.15) where −3  1  (1 − D)2 − C 2 + (1 + D)2 − C 2 2k± = 1− C     × 2 (C + D) 1 − (C − D)2 + (C − D) 1 − (C + D)2 ,    × ±4C C 2 − D 2 + (1 + C)2 − D 2 + (1 − C)2 − D 2 . (9.4.16) This can now take its place as a standard integration alongside the result for I (A, B). If the values of C and D given in (9.4.9) are substituted in the above, the result obtained for W S (αs , ws ) is precisely the same as that given by Delves and Joyce [15]. A further consequence of the relationship between I (A, B) and J (C, D) is that a connection between W F (α f , w f ) and W S (αs , ws ) may be found. Thus 2αs  W F (α f , w f ) = −  W S (αs , ws ), (9.4.17) 2 2 ws − (2 + αs ) + ws2 − (2 − αs )2 with wf = −

 ' ' 1 (ws2 − 4 − αs2 ) + ws2 − αs2 ws2 − (2 + αs )2 4αs  '  ' ' + ws2 − (2 − αs )2 + ws2 − (2 + αs )2 ws2 − (2 − αs )2 (9.4.18)

and

 ' ' 1 2 2 2 2 (ws − 4 − αs ) − ws − αs αf = − ws2 − (2 + αs )2 4αs  '  ' ' 2 2 2 2 2 2 + ws − (2 − αs ) + ws − (2 + αs ) ws − (2 − αs ) . (9.4.19)

310

Seventy years of the Watson integrals

The following inverse connection formula can also be found:   α f (w f + 2 − α f )(w f − 2 − α f ) 1/2 ws W S (αs , ws ) = w f W F (α f , w f ), w f (α f w f + 1) (9.4.20) where α f w f (w f + 2 − α f )(w f − 2 − α f ) wf + αf . (9.4.21) , αs = ws2 = − 2 αfwf + 1 (α f w f + 1) Thus we require either the result for W F (α f , w f ) or the result for W S (αs , ws ) for the other to be determined. All is explained in detail in a paper of Joyce et al. [37]. The ‘final problem’ yet to be solved is  π π π d x d y dz 1 . (9.4.22) W S (αs , βs , ws ) = 3 π 0 0 0 ws − αs cos x − βs cos y − cos z The latter might be referred to as the Watson integral for the doubly anisotropic cubic lattice or the Watson integral for an orthorhombic lattice. The difficulties encountered so far in attempted solutions of this three-parameter problem seem insuperable. Even attempts at reducing it to a Montroll-type two-parameter exercise, namely  π π π d x d y dz 1 , W S (1 + αs + βs ) = 3 π 0 0 0 1 + αs + βs − αs cos x − βs cos y − cos z (9.4.23) have not yielded any success.

9.5 The Green’s function of the simple cubic lattice Recently some significant progress has been made in finding exact product forms for certain forms of  π π π cos lx cos my cos nz 1 d x d y dz. G(l, m, n; αs , ws ) = 3 π 0 0 0 ws − αs cos x − cos y − cos z (9.5.1) A series of papers by Joyce and Delves [35, 36] and Delves and Joyce [16, 17] includes extensive detail in the presentation of results, which are briefly summarized here. These new results depend on a beautiful extrapolation of Iwata’s result for I (A, B). To see how this extrapolation is best presented, consider

 π  1 1 √ ,2 1 π 2 2 ; A + B cos x d x. K ( A + B cos x) d x = I (A, B) = 2 2 F1 π 0 1 π 0 (9.5.2) This may be taken to be a special case of    t, 1 − t 1 π In (t; A, B) = ; A + B cos x cos nx d x, (9.5.3) 2 F1 1 π 0

9.5 The Green’s function of the simple cubic lattice

311

where clearly Iwata’s I (A, B) is I0 ( 12 ; A, B). Now, Delves and Joyce [17] found a closed form for In (t; A, B), namely  2  √ (t)n (1 − t)n 1 √ In (t; A, B) = 1− A− B− 1− A+ B 2B (n!)2     t, 1 − t t, 1 − t ; θ+ 2 F1 ; θ− , (9.5.4) × 2 F1 n+1 n+1 where (t)n is the Pochhammer symbol, given by (t + n)/ (n), and   2θ± = 1 ± A2 − B 2 − (1 − A)2 − B 2 .

(9.5.5)

The remarkable expression (9.5.4) was first obtained using a Lie-group addition formula applicable to 2 F1 first obtained by Miller [45]. It was later confirmed by generalizing Iwata’s original approach. For some particular choices of (l, m, n) the Fourier transform of the Green’s function may be reduced to an elliptic integral. Some of these elliptic integrals correspond to Miller’s form, and the Green’s function can be evaluated in a few pages of hand calculation. For example, W S (2n, n, n; αs , ws ) was shown to be given by 1/2  1 W S (2n, n, n; αs , ws ) = ws2 + 4 − αs2

 1 3 2 − α 2 + α 2 cos x 2w , 1 π s s s 4 4 ;8 × cos nx d x. 2 F1 π 0 1 (ws2 + 4 − αs2 )2 (9.5.6) (It is noteworthy that in obtaining (9.5.6) the authors met a complex trigonometric integral whose solution as an elliptic integral was found in Jacobi [30]. An alternative procedure originating in Cayley [8, 9] was combined with an 2 F1 transformation of Goursat [25]. When stuck go to the experts.) Equation (9.5.6) is clearly of the form (9.5.4) and thus the following closed form is obtained:     3 1/2 1  4 n 4 n 1 W S (2n, n, n; αs , ws ) = ws2 + 4 − αs2 (n!)2 ' 2 2n ' 1 × ws2 − (2 − αs )2 − ws2 − (2 + αs )2 8αs



1 3 1 3 ,4 ,4 4 4 ; η+ 2 F1 ; η− , × 2 F1 (9.5.7) n+1 n+1     where 14 and 34 are Pochhammer symbols and n n '  ws ±16 2η± = 1 + 2 ws2 − αs2 (ws + 4 − αs2 )2 ' '  (9.5.8) − (ws2 − 4 − αs2 ) ws2 − (2 − α)2 ws2 − (2 + α)2 .

312

Seventy years of the Watson integrals

Similarly, 1 ws W S (n, n, n; 1, ws ) = π gives the closed form ws W S (n, n, n; 1, ws ) =

with





π 2 F1 0

27 (ws + cos x) cos nx d x ; 1 4ws3 (9.5.9)

1 2 3, 3

   

3n   ' 1 2−9 − w w s s 3



1 2 1 2 ,3 ,3 3 3 ; ξ+ 2 F1 ; ξ− , × 2 F1 n+1 n+1 1 3 n

2 3 n (n!)2

) )

1 9 1 2 2 ξ± (ws ) = 4ws + (9 − 4ws ) 1 − 2 ± 27 1 − 2 . 8ws2 ws ws

(9.5.10)

(9.5.11)

Interestingly, if in (9.5.7) and (9.5.10) n is made zero and ws is set equal to 3, two new formulae for the original Watson integral W S may be obtained. Along with (9.1.9) we thus have 

2 1 1 √ √ 2 √ √ √ √ , W S = (18 + 12 2 − 10 3 − 7 6) 2 F1 2 2 ; {(2 − 3)( 3 − 2)} 1

2

2   1 3 1 3 √ , , 1 1 8 8;1 4 4 ; 1 ( 2 − 1)2 = = 2 F1 2 F1 2 1 9 2 1 6

  √ √ 2 1 2 √ 2 2 ,3 1 3 − 1   1   11 2 3 ; 4 (2 − 2)  24  24 = = . 2 F1 3 1 96π 3 (9.5.12) We believe it was Littlewood who once remarked ‘All equations are identities.’ Does (9.5.12) support or give the lie to this statement?

9.6 Generalizations and recent manifestations of Watson integrals An obvious generalization of WIs is to higher dimensions. A recent survey by Guttmann [26] is an excellent source of information on these objects. Guttmann notes that for two-dimensional lattices WIs are given in terms of a single K . For three-dimensional lattices the solutions appear as products of two complete elliptic integrals K (k+ )K (k− ). It appears that this occurs because the underlying third-order ordinary differential equation (ODE) obeyed by these lattices has the almost-magical Appell reduction property, allowing their solution to be expressed in terms of an associated second-order ODE. For four-dimensional lattices it appears that the underlying ODEs are all of the Calabi–Yau type. This is true for all five-dimensional lattices as well, except for the five-dimensional FCC lattice.

9.6 Generalizations and recent manifestations of Watson integrals

313

Let us consider the four-dimensional simple cubic lattice in more detail. We have  π π π π d x d y dz dt 1 W S,4d (ws ) = 4 . (9.6.1) π 0 0 0 0 ws − cos x − cos y − cos z − cos t Its P-form is

 π π π π d x d y dz dt 1 4 π 0 0 0 0 1 − (u/4)u (cos x + cos y + cos z + cos t)   4 4 = WS . (9.6.2) u u

PS,4d (u) =

It has been shown that the d-dimensional diamond (D) lattice is simply related to the (d + 1)-dimensional hypercubic lattice via an Abel transform – see Guttmann and Prellberg [27], and Glasser and Montaldi [22]. The D lattice has not been previously discussed, since in three dimensions the analysis involved is identical to that in the FCC case. In terms of their respective P functions we have     4 3 1 1 PD,3d = PF,3d . (9.6.3) 4u + 4 1+u 4u u As can be seen, when u = 3 the two are identical apart from a numerical factor. Since everything about the three-dimensional diamond lattice is known, this suggests an interesting approach to the four-dimensional simple cubic lattice. Indeed it may be shown that the four-dimensional simple cubic P form may be expressed as a single integral (Guttmann [26]) namely  8 1 K (k+ )K (k− ) dt PS,4d (u) = 3 √ π 0 1 − t2 where k± =

1 2

  ± 14 u 2 t 2 4 − u 2 t 2 − 14 (2 − u 2 t 2 ) 1 − u 2 t 2 .

(9.6.4)

A single integral for W S,4d (ws ) was obtained by Zucker by doing two integrals following Iwata and a third integral following Joyce and Zucker [39], leading to  π 2 2  K (k ) p (1 − p 2 )(1 − 9 p 2 ) 8 dt, (9.6.5) W S,4d (ws ) = 3 π 0 (1 − p)3 (1 + 3 p) where γ = ws − cos t,

p = 2

γ− γ+

 

γ2 − 1 γ2 − 9

,

and

k2 =

16 p 3 . (1 − p)3 (1 + p)

Both (9.6.4) and (9.6.5) easily provide numerical values for any ws and u, but one is no nearer to a closer-form evaluation.

314

Seventy years of the Watson integrals

It has already been noted, in Section 9.2 that Bessel functions are connected to WIs. For example, it is simple to express W S as a single integral as follows. First, put  ∞ π  π  π 1 WS = 3 et (−3+cos x+cos y+cos z) d x d y dz dt; (9.6.6) π 0 0 0 0 since 1 π



π

et cos x d x = I0 (t),

(9.6.7)

0

we then have

 WS = 0

∞

e−3t I03 (t) dt,

(9.6.8)

where I0 (t) is a modified Bessel function of the first kind. Now, in a recent investigation, Bailey et al. [2] studied integrals of the form  ∞  ∞ k n t K 0 (t) dt, and tn,2k+1 = t 2k+1 I02 (t)K 0n−2 (t) dt cn,k = 0

0

(9.6.9) amongst many others. Here K 0 is a modified Bessel function of the second kind and should not be confused with a complete elliptic integral of the first kind. These integrals arose from certain Feynman diagrams and connections with WIs in three dimensions have been found, though some are still conjectures. It has been established that  ∞ π3 K 03 (t) dt = (9.6.10) W F (3) c3,0 = 2 0 and conjectured that 

∞

t5,1 = 0

?

t I02 (t)K 03 (t) dt = π 2 W F (15).

Also, Broadhurst [7] has proved that  ∞ π2 W F (3). t Io2 (t)K 02 (t)K 0 (2t) dt = 12 0

(9.6.11)

(9.6.12)

This may just be the tip of an iceberg and possibly many other similar results remain to be discovered. Another appearance of WIs is in the field of Mahler measures. The connection arises since the integrals may be expressed as a derivative with respect to the appropriate complex variable w of a logarithmic form. For example,    π π π 1 d log(w − cos x − cos y − cos z) d x d y dz . W S (ws ) = s dws π 3 0 0 0 (9.6.13)

9.7 Commentary: Watson integrals and localized vibrations

315

These logarithmic integrals are involved in the calculation of the total number of spanning trees on a hypercubic lattice (Rosengren [53]) and in the theory of collapsing branched polymers (Madras et al. [41]). A rapid way of evaluating these integrals in all dimensions was devised by Joyce and Zucker [38]. The definition of a Mahler measure m of an n-variable polynomial P(z 1 , . . . z n ) is (here P is not to be confused with a modified WI)  ∞  ∞   m [P(z 1 , . . . , z n )] = ... log |P e2πiθ1 , . . . e2πiθn | dθ1 · · · dθn . 0

0

(9.6.14) It was noted by Rogers [52] that two Mahler measures in which he was interested, namely   g1 (u) = m u + x + x −1 + y + y −1 + z + z −1 and g2 (u) = m − u + 4 + (x + x −1 )(y + y −1 )

+ (y + y −1 )(z + z −1 ) + (z + z −1 )(x + x −1 )

(9.6.15)

were closely related to the SC and FCC versions of WIs. This came as a surprise to Rogers as much as it did to some authors who were working on WIs. The appearance of WIs in these unexpected places shows their ubiquitous nature, and this seems an appropriate juncture to end this review.

9.7 Commentary: Watson integrals and localized vibrations (1) One way of thinking about WIs is that they provide a connection between lattice sums, periodic Green’s functions, and single-source Green’s functions. Lattice sums themselves can be regarded as Green’s functions. If we go back to Chapter 3, the Green’s function  of Section 3.3 was defined as a lattice sum for the generalized Poisson equation, with a set of sources at the points S = R p : ⎡ ⎤   s/2  = i s eik·S = i s (ab) ⎣ δ(S − R p ) − 1⎦ . (9.7.1) k=0

Rp

We can convert  from a multiple-source Green’s function to a single-source Green’s function by multiplying it by a suitable weight function and integrating over the unit cell of S. We use the property that  (S)e−ik·Rq d 2 S = 0 unless S = Rq . (9.7.2) UC

The single source chosen is then at S = Rq . A set of sources can be picked out using other weight functions. The Watson integral (9.5.1) is a localized Green’s function of this type.

316

Seventy years of the Watson integrals

(2) A recent article by Colquitt et al. [11] treats a two-dimensional vibrational problem in this fashion. In the case of N defects in an infinite square lattice, the Fourier-transformed equation for vibrations of angular frequency ω in the lattice is written as (ω2 − 4 + 2 cos ξ1 + 2 cos ξ2 )u(ξ ) =

N −1 

u p,0 e−i pξ1 .

(9.7.3)

p=0

This equation has oscillating solutions in ω2 < 8, but for ω2 > 8 the solution decays exponentially with increasing distance from the defect region. The interest here is in the behaviour of the solution in the exponentially decaying or stop-band frequency region. The field is written in terms of linear combinations of N shifted Green’s matrices, defined as  π π cos (n 1 − p)ξ1 cos n 2 ξ2 1 dξ1 dξ2 . (9.7.4) g(n, p; ω) = 2 2 π 0 0 ω − 4 + 2 cos ξ1 + 2 cos ξ2 These of course are the doubly periodic form of (9.5.1). Letting a = ω2 /2 − 2 + cos ξ2 , the Green’s matrix may be reduced to a single integral:  π √ 2 1 ( a − 1 − a)|n 1 − p| g(n, p; ω) = dξ2 . (9.7.5) √ 2π 0 a2 − 1 An alternative representation is in a Bessel-function product form, with two elements rather than the three occurring in (9.2.12). Also, instead of the oscillating function J0 we have the modified function I0 , given that we are assuming ω lies in the band gap:  (−1)n 1 − p+n 2 ∞ In 1 − p (x)In 2 (x) d x. (9.7.6) g(n, p; ω) = 2 0 One way of obtaining a shifted WI is to start with the elliptic integral form for the unshifted WI,   1 4 g(0, 0, 0; ω) = , (9.7.7) K aπ a2 and use repeated integration by parts together with the recurrence relation In (x) =  (x) − I 2In−1 n−2 (x). Colquitt et al. [11] also give a representation for g(n, p; ω) in terms of the hypergeometric function 4 F3 .

9.8 Commentary: Variations on W S As recounted already in this chapter, much effort has been put into evaluating the integral  π π π d x d y dz 1 (9.8.1) W = 3 π 0 0 0 3 − cos x − cos y − cos z

9.8 Commentary: Variations on W S and its generalization W (w) := W S (w) =

1 π3



π

0

 0

π



π 0

317

d x d y dz . w − cos x − cos y − cos z

(9.8.2)

For amusement (and reference) we list here solutions of W (w), in terms of singular values of complete elliptic integrals of the first kind, K (k 2 ), which have been discovered by various investigators. As in Section 1.12 we use √ b( p) = β( p) tan pπ

where

β( p) =

 2 ( p) (2 p)

and give W (w) for various values of w along the real axis.   2 1 b √ 3 W (0) = −21/3 3 i, 2 12π   2 1 √ b 8 , W (1) = (1 − 2i) 16π 2   1 b2 20 √ , W ( 5) = 21/5 (51/4 − 5−1/4 i) 80π 2   1 b2 24 2/3 , W (3) = 2 96π 2     b2 13 √ 3√ W 6 = 25/6 3 , 2 48π 2   b2 18 W (5) = , 48π 2       1 11 17 b 120 b 120 b 120 W (9) = 213/15 , % 7 & 480π 2 b 120     √ ⎡ b 1 b 25 ⎤2 5/21 168 168 7 2   ⎦ , W (15) = 5 2 2 ⎣ 37 2 3 7 πb 168

        √ ⎡ b 1 b 25 b 35 b 49 ⎤2 2/3 312 312 312 312 13 2       ⎦ , W (51) = 5 2 2 ⎣ 13 23 61 2 3 13 π b 312 b 312 b 312 √ 22/3 3 W (99) =  816                  1 25 35 41 49 59 65 83 91 b 408 b 408 b 408 b 408 b 408 b 408 b 408 b 408 b 408   % &           × . 5 7 13 29 31 47 79 b 408 b 408 b 408 b 408 b 408 b 408 π 2 b 408

318

Seventy years of the Watson integrals

Another set of integrals which may also be solved in terms of singular values is  π π π 1 α d x d y dz J (α) := 3 . (9.8.3) π 0 0 0 2α + 1 − α cos x − α cos y − cos z The general solution of this may be shown to be (1 − 2k − k 2 )2 . 8k(1 − k 2 ) √ For some values of α the value of k found is such that K  (k 2 ) = N K (k 2 ), where N is an integer. In this case the solution to (9.8.3) is (1 − 2k − k 2 )K (k 2 )K  (k 2 )

J (α) =

√

where

α :=



K (k 2 [N ]) N (1 − 2k[N ] − k [N ]) π 2

2 ,

and we thus obtain solutions in terms of singular values (see appendix section A.3). Thus, for α = 1, 4, 24, 49, 4900 we have N = 6, 10, 18, 22, 58 and hence ⎡   ⎤2 1  π π π 2/3 2 ⎣ b 24 ⎦ d x d y dz 1 = , J (1) = 3 96 π π 0 0 0 3 − cos x − cos y − cos z  π π π 1 4 d x d y dz J (4) = 3 π 0 0 0 9 − 4 cos x − 4 cos y − cos z ⎡     ⎤2 1 9 1 ⎣ b 40 b 40 ⎦   = , 80 πb 3 8

J (24) = J (49) = = J (4900) = = ×

1 π3 1 π3

 

π



π



0

π



π



0

π

0 π

  √ ⎡ b 1 ⎤2 24 d x d y dz 6⎣ 8 ⎦ = , 49 − 24 cos x − 24 cos y − cos z 24 π

49 d x d y dz 0 0 0 99 − 49 cos x − 49 cos y − cos z    ⎤2 ⎡   1 9 19 3/22 b b 88 88 b 88 2 ⎣   % & ⎦ , 5 7 176 b 88 π b 88  π π π 1 4900 d x d y dz , 3 π 0 0 0 9801 − 4900 cos x − 4900 cos y − cos z 228/29 35 232 ⎡                 ⎤2 1 9 25 33 35 49 51 57 b 232 b 232 b 232 b 232 b 232 b 232 b 232 b 232 ⎣ ⎦ .   % &           5 7 13 23 45 53 b 232 b 232 b 18 b 232 b 232 b 232 π b 232

9.9 Commentary: Computer algebra

319

The value J (4900) is surely one of the most remarkable closed forms obtained in the subject.

9.9 Commentary: Computer algebra The comment in [5] that for some problems ‘current computer algebra systems are of little help’ is slightly outdated. Such is the progress made in the last 5–10 years that a modern computer algebra system such as Maple can now do an enormous amount of work previously done using pencil and paper. For instance, if one knows the recurrence satisfied by a sequence (say (9.2.16)) then the differential equation satisfied by its generating function (in our example, (9.2.18)) can be obtained effortlessly using the command rectodiffeq from the Maple package gfun. Moreover, for many sums depending on a parameter n, a recurrence in n can be automatically found (and provably certified) using the Wilf–Zeilberger algorithm; these sums include a large number of binomial sums and many hypergeometric identities. Analogously, the Almqvist–Zeilberger algorithm can produce differential equations in t for integrals that depend on a parameter t. A very readable treatment is given in Petkovsek et al. [48], and implementations of these algorithms can be downloaded for free. We return to the differential equation (9.2.18). In Maple 13 (and newer versions), if one simply applies the command dsolve to it then very quickly a solution in terms of Heun functions is returned. This is possible because this particular third-order differential equation is a symmetric square (that is, its solutions are the products of the solutions of a second-order differential equation), a fact which can be readily checked; Maple then tries a variety of tricks to solve the pertinent second-order differential equation, including the application of a number of algebraic transformations. For (9.4.3), we mention that the generating function is annihilated by a sixth-order differential operator, which can be written as a product of a fourthorder and a second-order differential operator. Factorizations of this sort can be painlessly accomplished using the DFactor command from the Maple package DETools. Moreover, it is no longer necessary to perform hypergeometric or gamma function calculations by hand (as in (9.5.6)), since computer algebra systems are substantially faster and more reliable and they can be ‘taught’ key values and transformations for these functions. Finally, we remark that oversights such as being out by a factor of 384π in (9.3.4) can be greatly reduced when one uses a computer algebra system. First, one may save all the main results (and their numerical values) on the same worksheet and therefore numerical discrepancies may be spotted at a glance. Transcription errors can also be reduced owing to commands which convert the content

320

Seventy years of the Watson integrals

of a worksheet into LATEX. Suppose, however, that a numerical discrepancy has crept in and one suspects to be out by a factor F from the true answer. If F is likely to be a product of well-known constants (such as fractions, surds, multiples or powers of π ), then one could either use the Maple command identify on F or run the PSLQ algorithm on the vector consisting of log F as well as the logs of any suspected constants – such as log 2, log 3, log π – in an attempt to recover F. It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never-satisfied man is so strange if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others. Carl Friedrich Gauss From an 1808 letter to Farkas Bolyai (father of Janos Bolyai)

References [1] P. Appell. Sur les transformations des équations différentielles linéaires. C. R. Acad. Sci. Paris, 91:211–214, 1880. [2] D. H. Bailey, J. M. Borwein, D. Broadhurst, and M. L. Glasser. Elliptic integral evaluations of Bessel moments and applications. J. Phys. A: Math. Theor., 41:205203, 2008. [3] W. N. Bailey. A reducible case of the fourth type of Appell’s hypergeometric functions of two variables. Quart. J. Math. Oxford, 4:305–308, 1933. [4] W. N. Bailey. Some infinite integrals involving Bessel functions. Proc. London Math. Soc., 40:37–48, 1935. [5] F. Bornemann, D. Laurie, S. Wagon, and J. Waldvogel. The SIAM 100-digit Challenge. SIAM, 2004. [6] J. M. Borwein and I. J. Zucker. Elliptic integral evaluation of the Gamma function at rational values of small denominator. IMA J. Numer. Anal., 12:519–526, 1992. [7] D. Broadhurst. Elliptic integral evaluation of a Bessel moment by contour integration of a lattice Green function. 2008. Preprint: arXiv:0801.0891v1. [8] A. Cayley. The Collected Mathematical Papers, vol. 1. Cambridge University Press, 1889. [9] A. Cayley. An Elementary Treatise on Elliptic Functions, 2nd edition. Deighton, Bell and Co., 1895. αβ x + · · · ein Quadrat von der [10] T. Clausen. Ueber die Fälle wenn die Reihe y = 1 + 1·γ   

Form y = 1 + α1·δβ γ x + · · · hat. J. Math., 3:89–95, 1828. [11] D. J. Colquitt, M. J. Nieves, I. S. Jones, A. B. Movchan, and N. V. Movchan. Localisation for an infinite line defect in an infinite square lattice. Preprint: arXiv: 1208.1871v2:1–24, 2012.

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[12] R. Courant. Über partielle Differenzengleichung. In Proc. Atti Congresso Internazionale Dei Matematici-Bologna, vol. 3, pp. 83–89, 1929. [13] R. Courant, K. Friedrichs, and H. Lewy. Über die partiellen Defferenzengleichung der mathematischen Physik. Math. Ann., 100:32–74, 1928. [14] H. Davies. Poisson’s partial differential equation. Quart. J. Math, 6:232–240, 1955. [15] R. T. Delves and G. S. Joyce. On the Green function for the anisotropic simple cubic lattice. Ann. Phys., 291:71–133, 2001. [16] R. T. Delves and G. S. Joyce. Exact product form for the anisotropic simple cubic lattice Green function. J. Phys. A: Math. Theor., 39:4119–4145, 2006. [17] R. T. Delves and G. S. Joyce. Derivation of exact product forms for the simple cubic lattice Green function using Fourier generating functions and Lie group identities. J. Phys. A: Math. Theor., 40:8329–8343, 2007. [18] C. Domb. On multiple returns in the random walk problem. Proc. Camb. Phil. Soc., 50:586–591, 1954. [19] P. G. Doyle and L. J. Snell. Random Walks and Electric Networks. Carus Mathematical Monographs, MAA, 1984. [20] R. J. Duffin. Discrete potential theory. Duke Math. J., 20:233–251, 1953. [21] M. L. Glasser. A Watson sum for a cubic lattice. J. Math. Phys., 13:1145, 1972. [22] M. L. Glasser and E. Montaldi. Staircase polygons and recurrent lattice walks. Phys. Rev. E, 48:2339–2342, 1993. [23] M. L. Glasser and V. E. Wood. A closed form evaluation of the elliptical integral. Math. Comput., 25:535–536, 1971. [24] M. L. Glasser and I. J. Zucker. Extended Watson integrals for the cubic lattices. Proc. Nat. Acad. Sci. USA, 74:1800–1801, 1977. [25] E. Goursat. Sur l’équation différentielle linéaire qui admet pour intégrale la série hypergéometrique. Ann. Sci. École Norm. Sup., 10:S3–S142, 1881. [26] A. J. Guttmann. Lattice Green functions in all dimensions. J. Phys. A: Math. Theor., 43:305205, 2010. [27] A. J. Guttmann and T. Prellberg. Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions. Phys. Rev. E, 47:R2233–R2236, 1993. [28] F. T. Hioe. A Green’s function for a cubic lattice. J. Math. Phys., 19:1064–1067, 1978. [29] G. Iwata. Evaluation of the Watson integral of a face-centred lattice. Nat. Sci. Report, Ochanomizu University, 20:13–18, 1969. [30] C. G. Jacobi. Gesammelte Werke, vol. 3. Chelsea, New York, 1969. [31] G. S. Joyce. Lattice Green function for the anisotropic face centred cubic lattice. J. Phys. C, 4:L53–L56, 1971. [32] G. S. Joyce. On the simple cubic lattice Green function. Phil. Trans. Roy. Soc. London, A273:583–610, 1973. [33] G. S. Joyce. On the cubic lattice Green functions. Proc. Roy. Soc. London, A455:463– 477, 1994. [34] G. S. Joyce. On the cubic modular transformation and the cubic lattice Green functions. J. Math. A: Math. Gen., 31:5105–5115, 1998. [35] G. S. Joyce and R. T. Delves. Exact product forms for the simple cubic lattice Green functions: I. J. Phys. A: Math. Gen., 37:3645–3671, 2004a.

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[36] G. S. Joyce and R. T. Delves. Exact product forms for the simple cubic lattice Green functions: II. J. Phys. A: Math. Gen., 37:5417–5447, 2004b. [37] G. S. Joyce, R. T. Delves, and I. J. Zucker. Exact evaluation for the anisotropic facecentred and simple cubic lattices. J. Phys. A: Math. Gen., 36:8661–8672, 2003. [38] G. S. Joyce and I. J. Zucker. Evaluation of the Watson integral and associated logarithmic integral for the d-dimensional hypercubic lattice. J. Phys. A, 34:7349–7354, 2001. [39] G. S. Joyce and I. J. Zucker. On the evaluation of generalized Watson integrals. Proc. AMS, 133:71–81, 2004. [40] E. E. Kummer. Uber die hypergeometrische Reihe. J. Reine Angew. Math., 15:39–83, 127–172, 1836. [41] N. Madras, C. E. Soteros, S. G. Whittington, et al. The free energy of a collapsing branched polymer. J. Phys. A: Math. Gen., 23:5327–5350, 1990. [42] A. A. Maradudin, E. W. Montroll, G. H. Weiss, R. Herman, and W. H. Miles. Green’s Functions for Monatomic Simple Cubic Lattices. Acadèmie Royale de Belgique, 1960. [43] W. H. McCrea. A problem on random paths. Math. Gazette, 20:311–317, 1936. [44] W. H. McCrea and F. J. W. Whipple. Random paths in two and three dimensions. Proc. Roy. Soc. Edinburgh, 60:281–298, 1940. [45] J. Miller, Jr. Lie Theory and Special Functions. Academic Press, New York, 1968. [46] E. Montaldi. The evaluation of Green’s functions for cubic lattices, revisited. Lettera al Nuovo Cimento, 30:403–409, 1981. [47] E. W. Montroll. Theory of the vibration of simple cubic lattices with nearest neighbor interaction. In Proc. 3rd Berkeley Symp. on Math. Stats. and Probability, vol. 3, pp. 209–246, 1956. [48] M. Petkovsek, H. Wilf, and D. Zeilberger. A = B. A. K. Peters, Wellesley, 1996. [49] G. Pòlya. Uber eine Aufgabe der Wahrscheinlichkeitstheorie betreffend die Irrfahrt im Strassennetz. Math. Ann., 84:149–60, 1921. [50] S. Ramanujan. Modular equations and approximations to π . Quart. J. Math., 45:350– 372, 1914. [51] M. A. Rashid. Lattice Green’s functions for cubic lattices. J. Math. Phys., 21:2549– 2552, 1980. [52] M. Rogers. New 5 F4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/π . Ramanujan Journal, 18:327–340, 2009. [53] A. Rosengren. On the number of spanning trees for the 3D simple cubic lattice. J. Phys. A: Math. Gen., 20:L923–L927, 1987. [54] A. Selberg and S. Chowla. On Epstein’s zeta-function. J. Reine Angew. Math., 227:86–110, 1967. [55] I. Stewart. How to Cut a Cake: And Other Mathematical Conundrums. Oxford University Press, New York, 2006. [56] M. Tikson. Tabulation of an integral arising in the theory of cooperative phenomena. J. Res. Natl. Bur. Stds., 50:177–178, 1953. [57] W. F. van Pepye. Zür Theorie der magnetischen anisotropic Kubischer Kristalle beim absoluten Nullpunkt. Physica, 5:465–82, 1938.

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[58] G. N. Watson. The expansion of products of hypergeometric functions. Quart. J. Math., 39:27–51, 1908. [59] G. N. Watson. Three triple integrals. Quart. J. Math. Oxford, 10:266–276, 1939. [60] E. T. Whittaker and G. N. Watson. A Course of Modern Analysis, 4th edition. Cambridge University Press, 1946. [61] I. J. Zucker. The evaluation in terms of -functions of the periods of elliptic curves admitting complex multiplication. Math. Proc. Camb. Phil. Soc., 82:111–118, 1977.

Appendix

A.1 Tables of modular equations Table A.1 Modular equations of order 3 and dimensions 2 and 4, each forming a set of four elements. Ramanujan

Classical

Lambert series

4qψ(q 2 )ψ(q 6 )

θ2 θ2 (q 3 )

4

φ(q)φ(q 3 ) − 1

θ3 θ3 (q 3 ) − 1

2

∞ χ n  −6a q 2n 1−q 1 ∞  χ −3 q n

1 + (−q)n

1

φ(−q)φ(−q 3 ) − 1

θ4 θ4 (q 3 ) − 1

−2

∞ χ n  −6b q 1 + qn

Mellin transform 4(1 − 2−2s )L 1 L −3 2(1 + 21−2s )L 1 L −3 −2(1 − 22−2s )L 1 L −3

1

4qψ(−q 2 )ψ(−q 6 )

θ5 θ5 (q 3 )

16qψ 2 ψ 2 (q 3 )

θ22 (q 1/2 )θ22 (q 3/2 )

φ 2 (q)φ 2 (q 3 ) − 1

θ32 θ32 (q 3 ) − 1

φ 2 (−q)φ 2 (−q 3 ) − 1

θ42 θ42 (q 3 ) − 1

16qψ 2 (−q)ψ 2 (−q 3 )

θ52 (q 1/2 )θ52 (q 3/2 )

∞ χ n  12 q 2n 1+q 0 ∞ χ n  3a nq 16 2n 1−q 1  ∞  χ 3a 4nq 4n χ 6a nq n 4 + 1 − qn 1 − q 4n 1   ∞  χ 3a 4nq 4n χ 6a nq n −4 − 1 + qn 1 − q 4n

4

4L −4 L 12 16(1 − 2−s )(1 − 31−s )L 1 (s)L 1 (s − 1) 4(1 − 21−s + 22−2s )(1 − 31−s )L 1 (s)L 1 (s − 1) −4(1 − 22−s )(1 − 31−s )L 1 (s)L 1 (s − 1)

1

16

∞ χ χ n  3a 2b nq 2n 1−q 1

16(1 − 2−s )(1 − 22−s )(1 − 31−s )L 1 (s)L 1 (s − 1)

326

Appendix

Table A.2 Modular equations of order 3 and dimension 2 forming a set of eight elements Ramanujan ψ 3 (q) −1 ψ(q 3 ) φ 3 (q) −1 φ(q 3 ) φ 3 (−q) −1 φ(−q 3 ) ψ 3 (−q) −1 ψ(−q 3 ) ψ 3 (q 3 ) −1 ψ(q)

Classical   3 1/2 1 θ2 q % & −1 4 θ2 q 3/2 θ33 (q) θ3 (q 3 ) θ43 (q) θ4 (q 3 )

θ33 (q 3 )

φ 3 (−q 3 ) −1 φ(−q)

θ43 (q 3 )

ψ(−q)

−1

θ3 (q)

3

∞ χ n  −6a q 1 − qn

Mellin transform 3(1 + 2−s )L 1 L −3

1

∞ χ n  −6b q 6 1 + (−q)n

6(1 + 21−s − 21−2s ))L 1 L −3

1

−1

  3 1/2 1 θ5 q % & −1 4 θ5 q 3/2   3 3/2 1 θ2 q % & −1 4 θ2 q 1/2

φ 3 (q 3 ) −1 φ(q)

ψ 3 (−q 3 )

−1

Lambert series

−1

−6

∞ χ n  −3 q 1

−3

∞ χ n  −6a q n 1+q

θ4 (q)   3 3/2 1 θ5 q % & −1 4 θ5 q 1/2

−6(1 − 21−s )L 1 L −3 −3(1 + 2−s )(1 − 21−s )L 1 L −3

1

∞ χ n  −3 q 1 − q 2n

(1 − 2−s )L 1 L −3

1

−2

∞  1

−1

1 + qn

2

χ −3 q n 1 − (−q)n

∞ χ n  −6b q 1 − qn

−2(1 − 21−s − 21−2s )L 1 L −3 2(1 + 21−s )L 1 L −3

1

∞ χ n  −6b q 2n 1−q 1

(1 − 2−s )(1 + 21−s )L 1 L −3

Table A.3 Modular equations of order 3 and dimension 6 forming a set of eight elements; A = L 1 (s)L −3 (s − 2), B = L 1 (s − 2)L −3 (s) Ramanujan

Classical

Lambert series

  64 qψ 5 (q)ψ(q 3 ) − 9q 2 ψ(q)ψ 5 (q 3 )

θ25 (q 1/2 )θ2 (q 3/2 ) − 9θ2 (q 1/2 )θ25 (q 3/2 )

64

∞ χ 2 n  −3 n q 1 − q 2n

Mellin transform 64(1 − 2−s )A

1

9φ(q)φ 5 (q 3 ) − φ 5 (q)φ(q 3 ) − 8

9θ3 θ35 (q 3 ) − θ35 θ3 (q 3 ) − 8

8

∞ χ 2 n  −3 n q

1 − (−q)n

1

9φ(−q)φ 5 (−q 3 ) − φ 5 (−q)φ(−q 3 ) − 8   64 qψ 5 (−q)ψ(−q 3 ) + 9q 2 ψ(−q)ψ 5 (−q 3 )

9θ4 θ45 (q 3 ) − θ45 θ4 (q 3 ) − 8 θ55 (q 1/2 )θ5 (q 3/2 ) + 9θ5 (q 1/2 )θ55 (q 3/2 )

  64 qψ 5 (q)ψ(q 3 ) − q 2 ψ(q)ψ 5 (q 3 )

θ25 (q 1/2 )θ2 (q 3/2 ) − θ2 (q 1/2 )θ25 (q 3/2 )

φ 5 (q)φ(q 3 ) − φ(q)φ 5 (q 3 )

θ35 θ3 (q 3 ) − θ3 θ35 (q 3 )

φ(−q)φ 5 (−q 3 ) − φ 5 (−q)φ(−q 3 )

θ4 θ45 (q 3 ) − θ45 θ4 (q 3 )

−8

64

64

∞ χ 2 n  −6b n q 1 − qn

1 ∞ χ  1 ∞  1

  64 qψ 5 (−q)ψ(−q 3 ) + q 2 ψ(−q)ψ 5 (−q 3 )

θ55 (q 1/2 )θ5 (q 3/2 ) + θ5 (q 1/2 )θ55 (q 3/2 )

2 n −6b n q 1 − q 2n

n 2n χ −6a q + q (1 − q n )3

∞ 

2n n χ −3 q − (−q)

3 1 + (−q)n 1 ∞ n 2n  χ −3 q − q 8 n (1 + q )3

8

64

1 ∞  1

n 2n χ −6a q − q n (1 − q )3

8(1 + 21−s )(1 − 22−s ))A −8(1 + 23−s )A 64(1 − 2−s )(1 + 23−s )A 64(1 + 2−s )B 8(1 − 21−s )(1 + 22−s )B 8(1 − 23−s )B 64(1 + 2−s )(1 − 23−s )B

Table A.4 Modular equations of order 5 and dimension 4 forming a set of eight elements; C = L 1 (s)L 5 (s − 1), D = L 1 (s − 1)L 5 (s) Ramanujan

Classical

  16 qψ 3 (q)ψ(q 5 ) − 5q 2 ψ(q)ψ 3 (q 5 )

θ23 (q 1/2 )θ2 (q 5/2 ) − 5θ2 (q 1/2 )θ23 (q 5/2 )

5φ(q)φ 3 (q 5 ) − φ 3 (q)φ(q 5 ) − 4

5θ3 θ33 (q 5 ) − θ33 θ3 (q 5 ) − 4

5φ(−q)φ 3 (−q 5 ) − φ 3 (−q)φ(−q 5 ) − 4

5θ4 θ43 (q 5 ) − θ43 θ4 (q 5 ) − 4

Lambert series ∞ χ n  5 nq 16 1 − q 2n 1 ∞  χ 5 nq n

4

1 − (−q)n

1

  16 qψ 3 (−q)ψ(−q 5 ) + 5q 2 ψ(−q)ψ 3 (−q 5 )   16 qψ 3 (q)ψ(q 5 ) − q 2 ψ(q)ψ 3 (q 5 ) φ 3 (q)φ(q 5 ) − φ(q)φ 3 (q 5 )

θ53 (q 1/2 )θ5 (q 5/2 ) + 5θ5 (q 1/2 )θ53 (q 5/2 ) θ23 (q 1/2 )θ2 (q 5/2 ) − θ2 (q 1/2 )θ23 (q 5/2 ) θ33 θ3 (q 5 ) − θ3 θ33 (q 5 )

φ(−q)φ 3 (−q 5 ) − φ 3 (−q)φ(−q 5 )

θ4 θ43 (q 5 ) − θ43 θ4 (q 5 )

  16 qψ 3 (−q)ψ(−q 5 ) + q 2 ψ(−q)ψ 3 (−q 5 )

θ53 (q 1/2 )θ5 (q 5/2 ) + θ5 (q 1/2 )θ53 (q 5/2 )

−4

16

16

4

4

∞ χ n  10b nq n 1−q

1 ∞ χ 

1 ∞ 

16

4(1 − 21−s − 22−2s ))C −4(1 + 22−s )C 16(1 − 2−s )(1 + 22−s )C

n 10a q (1 − q n )2

16(1 + 2−s )D

χ 10b q n [(1 + (−q)n ]2 χ 5qn (1 + q n )2

1 ∞ χ  1

16(1 − 2−s )C

n 10b nq 2n 1−q

1 ∞ χ 

1 ∞ 

Mellin transform

n 10a q n (1 + q )2

4(1 + 21−s − 22−2s ))D 4(1 − 22−s )D 16(1 + 2−s )(1 − 22−s )D

Table A.5 Modular equations of order 7, and mixed equations of orders 3, 5, and 15 Ramanujan 4qψ(q)ψ(q 7 ) φ(q)φ(q 7 ) − 1

Classical θ2 (q 1/2 )θ2 (q 7/2 ) θ3 θ3 (q 7 ) − 1

Lambert series 4

2

n −14a q 1 − qn

1 ∞ χ 

n −14b q 1 + (−q)n

1

φ(−q)φ(−q 7 ) − 1

Mellin transform

∞ χ 

∞ χ n  −7 q 1 + qn

θ4 θ4 (q 7 ) − 1

−2

θ5 (q 1/2 )θ5 (q 7/2 )

∞ χ n  −14a q 4 n 1+q

4(1 − 2−s )L 1 L −7 2(1 − 21−s + 21−2s )L 1 L −7 −2(1 − 21−s )L 1 L −7

1

4qψ(−q)ψ(−q 7 )   4 q 2 ψ(q)ψ(q 15 ) + qψ(q 3 )ψ(q 5 ) φ(q)φ(q 15 ) + φ(q 3 )φ(q 5 ) − 2

θ2 (q 1/2 )θ2 (q 15/2 ) + θ2 (q 3/2 )θ2 (q 5/2 ) θ3 θ3 (q 15 ) + θ3 (q 3 )θ3 (q 5 ) − 2

4

2

0 ∞ χ 

−30a q 1 − qn

1 ∞ χ 

θ4 θ4 (q 15 ) + θ4 (q 3 )θ4 (q 5 ) − 2

−2

−30b q

n

1 + (−q)n

1

φ(−q)φ(−q 15 ) + φ(−q 3 )φ(−q 5 ) − 2

n

∞ χ n  −15 q 1 + qn

4(1 − 2−s )(1 − 21−s )L 1 L −7 4(1 − 2−s )L 1 L −15 2(1 − 21−s + 21−2s ))L 1 L −15 −2(1 − 21−s )L 1 L −15

1

  4 q 2 ψ(−q)ψ(−q −15 ) − qψ(−q 3 )ψ(−q 5 )

θ5 (q 1/2 )θ5 (q 15/2 ) − θ5 (q 3/2 )θ5 (q 5/2 )

4

∞ χ n  −30a q 1 + qn 1

4(1 − 2−s )(1 − 21−s )L 1 L −15

A.2 Character table for Dirichlet L-series Table A.6 Character table for the Dirichlet L-series (see (1.4.1)). The column headings are values of the integer n. The vertical bars indicate the ends of number pattern sequences n=

1

χ1 χ 2a χ 2b χ −3 χ 3a χ −4 χ5 χ 6a χ −6a χ −6b χ −7 χ 10a χ 10b χ 12 χ −14a χ −14b χ −15 χ −30a χ −30b

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0| 1 0| 1 0| 1 0| 1 0| 1 0| 1 0| 1 0| 1 0| 1 0| 1 −1| 1 −1| 1 −1| 1 −1| 1 −1| 1 −1| 1 −1| 1 −1| 1 −1| 1 −1| 1 −1 0| 1 −1 0| 1 −1 0| 1 −1 0| 1 −1 0| 1 −1 0| 1 −1 1 1 0| 1 1 0| 1 1 0| 1 1 0| 1 1 0| 1 1 0| 1 1 1 0 −1 0| 1 0 −1 0| 1 0 −1 0| 1 0 −1 0| 1 0 −1 0| 1 −1 −1 1 0| 1 −1 −1 1 0| 1 −1 −1 1 0| 1 −1 −1 1 0| 1 0 0 0 1 0| 1 0 0 0 1 0| 1 0 0 0 1 0| 1 0 1 0 0 0 −1 0| 1 0 0 0 −1 0| 1 0 0 0 −1 0| 1 0 1 1 0 −1 −1 0| 1 1 0 −1 −1 0| 1 1 0 −1 −1 0| 1 1 1 1 −1 1 −1 −1 0| 1 1 −1 1 −1 −1 0| 1 1 −1 1 −1 −1 1 0 −1 0 0 0 −1 0 1 0| 1 0 −1 0 0 0 −1 0 1 0| 1 1 −1 −1 0 −1 −1 1 1 0| 1 1 −1 −1 0 −1 −1 1 1 0| 1 0 0 0 −1 0 −1 0 0 0 1 0| 1 0 0 0 −1 0 −1 0 1 0 −1 0 −1 0 0 0 1 0 1 0 −1 0| 1 0 −1 0 −1 0 1 −1 −1 −1 −1 1 0 −1 1 1 1 1 −1 0| 1 −1 −1 −1 −1 1 1 1 0 1 0 0 −1 1 0 0 −1 0 −1 −1 0| 1 1 0 1 0 1 0 0 0 0 0 −1 0 0 0 −1 0 −1 0 0 0 1 0 1 0 1 −1 0 −1 0 0 −1 −1 0 0 −1 0 −1 1 0 −1 1 0 1 0

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20 21

22

23

24 25

1 1 1 1 1 0| 1 0| 1 −1| 1 −1| 0| 1 −1 0| 0| 1 1 0| 1 0 −1 0| 1 −1 −1 1 0 0 1 0| 0 0 −1 0| 0 −1 −1 0| 0| 1 1 −1 1 0 −1 0 1 1 −1 −1 0 0 1 0| 0 0 1 0 0 −1 1 1 0 −1 1 0 0 0 1 0 0 1 1 0

26

27

28

29

30

1 1 1 1 1 1 1 0| 1 0| 1 0| 1 −1| 1 −1| 1 −1| 1 −1 0| 1 −1 0| 1 1 0| 1 1 0| 1 0 −1 0| 1 0 0| 1 −1 −1 1 0| 1 0 0 0 1 0| 1 0 0 0 −1 0| 1 1 0 −1 −1 0| 1 −1 −1 0| 1 1 0 0 −1 0 1 0| 0 −1 −1 1 1 0| 1 0 0 0 −1 0 1 0 −1 0| 1 0 1 1 −1 0| 1 −1 0 −1 0 −1 −1 0| 0 0 0 0 −1 0| 0 1 0 1 −1 0|

A.3 Values of K [N ] for all integer N from 1 to 100

331

A.3 Values of K [N ] for all integer N from 1 to 100 The relevant definitions are to be found in Section 1.12.

1 1 1 2 , k1 = ; b 4 4 2  √ 1/2   √  4+2 2 2 b 18 , k  2 = 2 2 − 1 , Q[2] = 2; K [2] = 16 √ √ 2+ 3 22/3 3  1  2 K [3] = b 3 , k3= ; 12 4 √ √ 2 2 + 2 1 2 b 4 , k  4 = 25/2 2−1 ; K [4] = 16  1/2 √ √ 1/4 3/5   5 − 2 1 + 2 2 × 53/8 2 + 5  1 , k52 = K [5] = b 20 , 40 2 W [5] = 3;  √ √ 1/2   √ 27/12 31/4 1 + 2 + 6 √ 2  √ 2 1 , k62 = 2− 3 , b 24 3− 2 K [6] = 48 √ 2 P[6] = 2−1     √ ⎡b 1 b 2 ⎤ √ 7 7⎣ 7 8−3 7 2 ⎦   , k7 = ; K [7] = 14 16 b 3 K [1] =

7

K [8] =

  √ √ 1/2 4+6 2+8 1+ 2

 

1 8 , 32 √ 4  √  1 1+4 2 ; k82 + 2 = 2 2+1 k8  2 √  √ √  √   2− 3 3+1 2 − 31/4 31/4 2 K [9] = b 14 , k92 = , 24 2   √ 4 V [9] = 2 − 3 ;   √ √ √ 1/2 ⎡  1   9  ⎤ 21/2 5 2 + 6 5 + 2 10 b 40 b 40 ⎦, ⎣   K [10] = 80 b 18 √ 2 √ 4 2 = 10 − 3 2 − 1 , Q[10] = 3; k10

b

332

Appendix    √ 1/3  √ 1/3 1/2 661/2 4 + 19 + 3 33 + 19 − 3 33

K [11] =

132 ⎡         ⎤1/3 1 3 4 5 b 11 b 11 b 11 b 11 ⎦ ; ⎣   × 2 b 11  √ √ √ √ 4   6+ 2+4 22/3 3 √ 4 √ 2 = 3− 2 2−1 ; K [12] = b 13 , k12 96  √ 1/4 ⎡     ⎤ 1 9 b 52 133/8 × 210/13 5 13 + 18 b 52 ⎦, ⎣   K [13] = 3 104 b 52  √ 1 − 6 5 13 − 18 2 , W [13] = 11; k13 = 2   √ √ 29/14 × 75/8 √ 2 + 1 3 + 6 2 + 2 14 K [14] = 112 ⎡       ⎤1/2 ' 1/4 b 5 b 1 b 13 √ √ 56 8 56 ⎣ ⎦ ,   + 2 2 + 12 2 + 4 14 17 b 56 √ Q[14] = 1 + 2 2;  √ 1/2 ⎡  1   4  ⎤ b 15 b 15 61/2 5 + 5 ⎦, ⎣   K [15] = 5 60 b 15 2 k15 =

√ K [16] =

 √ 2 √ √ 2 √ 2  2− 3 3− 5 5− 3

2 + 21/4 32

128

2

  b

1 4

,

√

2 k16 = %

2−1

;

4

21/4 + 1

&8 ;

  1/4 1/2 √ √ 221/34 177/16 5 17 + 4 + 6 + 2 17 K [17] = 136 ⎡       ⎤1/2 1 9 b 68 b 68 b 13 68 ⎣ ⎦ ,   × 15 b 68 √ 17 + 5 17 W [17] = ; 2

A.3 Values of K [N ] for all integer N from 1 to 100

K [18] =

 √ √ 1/2 28 + 10 2 + 8 3

  b

1 8

,

 6 √ 4 √ 2 k18 = 2− 3 2−1 ,

48 √ √ 4 P[18] = 3− 2 ;    √ 1/3  √ 1/3 1/2 1/2 4 + 163 + 9 57 114 + 163 − 9 57

K [19] =

228 ⎡         % &   ⎤1/3 1 4 5 6 7 9 b 19 b 19 b 19 b 19 b 19 b 19 ⎦ ;       ×⎣ 2 3 8 b 19 b 19 b 19 1/2 √ 5−2 1+2 1 + k5 K [20] = K [5], k52 = , 2 2

(  √ 3  √  √  1 2 5+2 k20 + 2 = 2 3 62 + 11 5 + 32 5 2 + 5 5 ; k20 √  √  √ 1/4 211/21 213/8   2 2+ 3 3+ 7 + 7 K [21] = 168 ⎡       ⎤1/2 1 5 b 84 b 17 b 84 84 ⎦ ,   × ⎣ b 14  √ 2 √ 2  √ 3 3−2 7 ; V [21] = 8 − 3 7  √ √ √ 1/2 ⎡  1   9   19  ⎤ b 88 b 88 b 88 239/44 33 2 + 7 11 + 5 22 ⎦, ⎣   % & K [22] = 5 7 176 b 88 b 88

Q[22] = 6; √    √ 1/3  √ 1/3 1/6 65/6 23 K [23] = 20 + 6308 + 84 69 + 6308 − 84 69 276 ⎡               ⎤1/3 1 2 3 4 6 8 9 b 23 b 23 b 23 b 23 b 23 b 23 b 23 ⎦ ,   % &     ×⎣ 5 7 11 b 23 b b 23 b 10 23 23 ⎡   1/3 1/3 ⎤ √ √ 231 69 − 1181 231 69 + 1181 1 ⎣ ⎦; −22 − 5 +5 V [23] = 192 2 2

K [24] =

1+

√

3−1

 √  √ √ 1/2 2+1 3− 2 2

K [6],

333

334

Appendix 

 √  √  √ √ 1/2 1−u 2 = , u= 2+1 3+1 3− 2 , 1+u  √  √ √ 4  √  1 2 209 + 104 3 + 48 2 4 + 3 ; + 2 =2 2+ 3 k24 k24  2 % √ √ &2   3 − 2 × 51/4 5+2 5−2 2 b 14 , k25 , K [25] = = 20 2 W [25] = 47; ⎡           ⎤1/3 1 3 9 17 25 R 1/12 × 2233/312 × 135/12 ⎣ b 104 b 104 b 104 b 104 b 104 ⎦     , K [26] = 23 208T [26]1/8 b 18 b 104   √ 2 √ √ 1 R= 34 − 13 13 − 2 3 + 2 (4034 3   √ 2 √ √  1/3 √ √ 2 3− 2 + 3100 6 − 2105 13 − 457 78    √ √ √  1/3 × −4034 + 3100 6 + 2105 13 − 457 78 , 2 k24

   √ 1/3  √ 1/3 1 8 + 359 − 12 78 ; + 359 + 12 78 3 &2   % 2/3 2 +2 1 K [27] = b 16 , V [27] = + 25 × 21/3 − 20 × 22/3 ; 72 4 √ √ 8 + 3 2 + 14 K [7]; K [28] = 16 255/87 × 31/4 × 2911/24 K [29] = 696 ⎧  ⎤1/3 ⎡ √  ⎤1/3 ⎫ ⎡ √ ⎪ ⎪ ⎬ ⎨ 2 2 87 + 9 87 − 9 ⎣ ⎣ ⎦ ⎦ − × −1 + ⎪ ⎪ 9 9 ⎭ ⎩ ⎧

1/3 (  ⎨ 9 + √29 √  √ × + 369 + 70 29 + 12 6 27 + 5 29 ⎩ 2

1/3 ⎫3/4 (  ⎬ √ √  + 369 + 70 29 − 12 6 27 + 5 29 ⎭ Q[26] =

⎡           ⎤1/3 1 5 9 13 25 b 116 b 116 b 116 b 116 b 116 ⎦ ,   × ⎣ % 7 & 23 b 116 b 116    √ 1/3  √ 1/3 1 W [29] = 71 + 343601 − 2688 87 ; + 343601 + 2688 87 3

A.3 Values of K [N ] for all integer N from 1 to 100

335

√ √ √ √ √ 1/2 241/60 × 31/4 1/2  4 + 2 + 2 3 + 5 + 3 6 + 10 5 240 ⎡       ⎤1/2 1 11 17 b 120 b 120 b 120 ⎦ , × ⎣ % 7 & b 120 √ √ P[30] = ( 10 − 3)2 ( 5 − 2)2 ; √ 31 1/3 K [31] = R 62 ⎡         % &         ⎤1/3 1 2 4 5 7 8 9 14 b 31 b 31 b 31 b 31 b 31 b 31 b 10 b 31 31 b 31 ⎦ ,             × ⎣ 3 6 12 13 15 b 31 b 11 b b b b 31 31 31 31 31 ⎡  √ 1/3 ⎤ √ 1/3  187 + 9 93 187 − 9 93 1 ⎦; + R = ⎣5 + 3 2 2

K [30] =

1 + k8 K [8]; 2 37/66 √ √ √ √ × 333/8 2 (3 + 11)1/2 ( 3 + 1)3/2 (2 3 − 11)1/4 K [33] = 528 ⎡         ⎤1/2 1 17 25 29 b 132 b 132 b 132 b 132 ⎦ ,     × ⎣ 31 b 14 b 132 √ √ V [33] = (2 − 3)6 (10 − 3 11)2 ;  √

2123/136 × 173/8 9 − 4 33 17 − 131 K [34] = 272 T [34]1/8          ⎤1/2  ⎡ 1 9 19 25 33 b 136 b 136 b 136 b 136 b 136 ⎦ , ⎣       × 13 15 21 b 136 b 136 b 136 √ Q[34] = 3/2( 17 + 3); √ √ √ √ 1 K [35] = 2 + 2 5 + (146 + 91 5 − 3 21 + 6 105)1/3 2 1/2 √ √ √ +(146 + 91 5 + 3 21 + 6 105)1/3 ⎡           ⎤1/3 1 3 13 16 b 35 b 35 b 11 35 b 35 b 35 ⎦ ;     ×⎣ 2 b 15 b 35  √  √ √   31/4 2 1 + 2 + 3 + 31/4 b 14 ; K [36] = 48 K [32] =

336

Appendix ⎡          ⎤ 1 9 21 25 33 √ 1/2 b 148 b 148 b 148 b 148 b 148 22/37 × 373/8 ⎦,   % &     (6 + 37) ⎣ K [37] = 3 7 11 27 148 b 148 b 148 b 148 b 148 W [37] = 146;

R 1/12 × 2347/456 × 195/12 304T [38]1/8 ⎡             ⎤1/3 1 3 9 17 25 27 b 152 b 152 b 152 b 152 b 152 b 152 ⎦       × ⎣ 5 15 31 b 152 b 152 b 152

√ √ √ √ ( 2 + 1)3 2(2747 + 859 2) R=− 11 + 18 2 + + 2(7 − 5 2)S , 3 S  √ 1/3 √ √ √ , S = (358573 + 368494 2)( 2 + 1)4 + 66 57(32771 + 23166 2)       √ √ 1/3 1/3 1 + 1369 + 30 114 Q[38] = 16 + 1369 − 30 114 ; 3   1/4 ' √ √ √ ⎡    ⎤ 2 5 13 108 39 + 11 13 + 6(325 + 91 13 b 39 b 39 ⎦; ⎣ K [39] = %7& 468 b 39

K [38] =

 1 + k10 K [10]; 2 ' 1/8  √ 271/82 × 4115/32 R 1/8 K [41] = −8 + 32 + 5 41 328 ⎡               ⎤1/4 1 5 9 21 25 33 37 b 164 b 164 b 164 b 164 b 164 b 164 b 164 ⎦ ,       ×⎣ 23 31 39 b 164 b 164 b 164   1 v + v2 − 4 , R=  4 √ v = 5 41 + 32 '    √  √ √ × 33440 + 5162 41 + 32 211 + 35 41 × 10 + 2 41 , ' √ √  1 W [41] = 207 + 31 41 + 2(42281 + 6605 41) ; 4 √ √ √ √ √ 231/84 × 33/4 × 71/4 K [42] = (6 + 4 2 + 2 3 + 5 6 + 7 + 42)1/2 1008 ⎡    ⎤ 1 25 b 168 b 168 ⎣ ⎦,   × 37 b 168  √ 3 √ √ 5 − 21 (2 2 − 7)2 ; P[42] = 2

K [40] =

A.3 Values of K [N ] for all integer N from 1 to 100 ' K [43] =

337

√ √ 258(16 + (3547 + 276 129)1/3 + (3547 − 27 129)1/3 )

516                ⎤1/3 1 4 6 9 11 13 14 b 43 b 43 b 43 b 43 b 10 43 b 43 b 43 b 43 ⎢ ⎥         ⎢ ⎥ 16 17 21 ⎢ ⎥ b b b × b 15 43 43 43 43 ⎢ ⎥ × ⎢       % &          ⎥ ⎢ b 2 b 3 b 5 b 7 b 8 b 12 b 18 b 19 b 20 ⎥ ⎢ 43 43 43 43 43 43 43 43 43 ⎥ ⎣ ⎦ ⎡



 1 + k11 K [11] 2  √ √ 1+2 3+2 5 K [45] = K [5] 3  √ √

1/8 2177/184 × 233/8 9(6 2 − 7) − 14 36 2 − 50 K [46] = 368 T [46] ⎡               ⎤1/2 1 9 11 19 25 41 43 b 184 b 184 b 184 b 184 b 184 b 184 b 184 ⎦ ,   % &       ×⎣ 7 13 15 29 b 184 b 184 b 184 b 18 b 184 √ Q[46] = 3( 2 + 1)2 ; √ 47 [a(a + 1)]2/5 K [47] = 94 ⎡           % &       ⎤1/5 1 2 3 4 6 7 8 9 b 47 b 47 b 47 b 47 b 47 b 47 b 47 b 12 b 47 47 ⎥ ⎢           ⎢ ⎥ 14 16 17 18 21 ⎢ ⎥ × b 47 b 47 b 47 b 47 b 47 ⎢ ⎥ × ⎢                  ⎥ ⎢ b 5 b 10 b 11 b 13 b 15 b 19 b 20 b 22 b 23 ⎥ ⎢ 47 47 47 47 47 47 47 47 47 ⎥ ⎣ ⎦

K [44] =

where a is the only real root of the equation a 5 − a 3 − 2a 2 − 2a − 1 = 0: 1, s(1)1/5 + s(2)1/5 − [−s(3)]1/5 + s(4)1/5 , a= 5    √ s(n) = Re −650 + 80 −47) exp − 25 πin     √ + −975 + 15 −47 exp − 45 πin ;  √ √ √  √ √ 3/2 2−1 3− 2 6+ 2 1+ K [12]; K [48] = 2

338

Appendix

K [49] =

 √  √ √ 1/2 7 + 4 7 + 71/4 2 5 + 7

  b

28    √ √  W [49] = 8 + 3 7 16 + 3 7 ;

1 4

,

√ √ √ √ K [50] = 5 − 2 5 + 2 10 + 2 55 − 30 2 − 11 5 + 10 10 

√

1/3  √ √ √  + 3 30 29 − 20 2 − 11 5 + 8 10 (



√ √ √ + 2 55 − 30 2 − 11 5 + 10 10

1/3 ⎫ ⎬ K [2]  √ √ √  × , − 3 30 29 − 20 2 − 11 5 + 8 10 ⎭ 15     √ 1/3   √ 1/3 1 16 + 5 2279 − 84 6 ; + 5 2279 + 84 6 Q[50] = 3 √   √   √ 1/3  √ 1/3  1/2 35/6 102  K [51] = 2 1 + 17 + 497 + 151 17 + 623 + 151 17 612 ⎡             ⎤1/3 1 5 13 16 20 b 51 b 11 b 17 51 b 51 b 51 b 51 ⎦ ;   % &   ×⎣ 2 7 8 b 51 b 51 b 51 (

K [52] =

K [53] =

 1 + k13 K [13]; 2

 1/3  √ 1/3 1/3  √ − 2 3 159 − 10 −1 + 2 3 159 + 10

219/106 × 47711/24 636 R 1/4 ⎡                 ⎤1/3 1 9 13 17 25 29 37 49 b 212 b 212 b 212 b 212 b 212 b 212 b 212 b 212 ⎦ ,         ×⎣ % 7 & 11 15 43 47 b 212 b 212 b 212 b 212 b 212 √  √ R= 53 − 7 1661 − 217 53   √ 1/3 √ + 16 2 2077192 + 165 3 − 285355 53    √ √ 1/3 − 16 2 −2077192 + 165 3 + 285355 53 ,

  √ 1/3 1 505 + 8 266489 − 210 159 W [53] = 3   √ 1/3 ; + 8 266489 + 210 159

A.3 Values of K [N ] for all integer N from 1 to 100

339

 √    2  √ √  1/3 √ 2+1 9−5 3 K [54] = 2−1 3− 6+2 6 9   √ √ 1/3 √ + 6 6+6 3−9 2 K [6]; K [55] =

'   √ 1/4 23/4 × 57/8 × 111/4 √ 5+1 3+ 3+2 5 440   % &   2 7 b 17 b 55 b 55 55     ; × 3 23 b 55 b 55

 1 + k14 K [14]; 2 3/2  √ 1/4 √ 1/2  √ 225/38 × 573/8 √ 13 + 3 19 2 19 − 5 3 3+1 K [57] = 912 ⎡             ⎤1/2 1 25 29 41 49 53 b 228 b 228 b 228 b 228 b 228 b 228 ⎣ ⎦ , × % 7 &  43   55   1  b 228 b 228 b 4 b 228  √ 6  √ 2 170 − 39 19 ; V [57] = 2 − 3

K [56] =

K [58] =

√ √ 1/2 257/58 × 581/4  70 + 99 2 + 13 29 464 ⎡                ⎤ 1 9 25 33 35 49 51 57 b 232 b 232 b 232 b 232 b 232 b 232 b 232 b 232 ⎦,   % &           × ⎣ 5 7 13 23 45 53 b 232 b 232 b 232 b 18 b 232 b 232 b 232 Q[58] = 27;

K [59] =

√   1/3   √ √ 1/3 1/3 2 62/3 59 + 22/3 43 + 3 177 R −8 + 22/3 43 − 3 177 1416 ⎡         % &         ⎤1/9 1 3 4 5 7 9 15 16 b 59 b 59 b 59 b 59 b 12 b 59 b 59 59 b 59 b 59 ⎦           ×⎣ 2 6 8 11 b 59 b 59 b 59 b 10 b 59 59 ⎡                     ⎤1/9 17 19 20 21 22 25 27 28 29 b 59 b 59 b 59 b 59 b 59 b 59 b 26 59 b 59 b 59 b 59 ⎦ ,           ×⎣ 14 18 23 24 b 13 59 b 59 b 59 b 59 b 59

R = α + (β + γ )1/3 + (β − γ )1/3 ,    1/3 1/3  √ √ 1 4 + 22/3 43 − 3 177 , + 22/3 43 + 3 177 α= 9    1 129 + 3 × 22/3 (u − v) − 2 × 21/3 u 2 + v 2 , β= 243

340

Appendix

 √ 1/3  √ 1/3 u= 3 177 − 9 , v= 3 177 + 9 ,     √ 1/3 1/2 √ 1/3 1 − 4 44 − 3 177 ; 25 − 4 44 + 3 177 γ = 9 √  √  √  √ √  5+ 3 16 + 2 3 − 5 2 + 3 K [60] = K [15]; 32 2139/183 × 32/3 × 6111/24 K [61] = 1464   √ 1/3  √ 1/3 1/3 −5 + 2 6 183 + 62 − 2 6 183 − 62 × V [61]1/8 ⎡             ⎤1/3 1 5 9 13 25 41 b 244 b 244 b 244 b 244 b 244 b 244 ⎢ ⎥       ⎢ ⎥ 45 49 57 ⎢ ⎥ b 244 b 244 × b 244 ⎢ ⎥ ×⎢            ⎥ , ⎢ b 3 b 15 b 19 b 27 b 39 b 47 ⎥ ⎢ 244 244 244 244 244 244 ⎥ ⎣ ⎦     √ 1/3 √ 1/3 ; + 3 976247 + 144 183 W [61] = 296 + 3 976247−144 183 K [62] =

 1/8 2189/248 × 317/16 R 496 T [62] ⎡               ⎤1/4 1 3 9 11 25 27 33 b 248 b 248 b 248 b 248 b 248 b 248 b 248 ⎢ ⎥       ⎢ ⎥ 41 43 49 ⎢ ⎥ b 248 b 248 × b 248 ⎢ ⎥             ×⎢ ⎥ , ⎢ ⎥ 5 15 23 1 45 55 b 248 b 248 b 248 b 8 b 248 b 248 ⎢ ⎥ ⎣ ⎦ '  1/2 ' √ √ √ √ , R = 25 − 19 2 + 2 2(−13 + 10 2 − 10 2−16 2 + 8 1 + 4 2 √ Q[62] = √

K [63] = K [64] =

2+1

2

'



2  √ 1/2 3 + 9 + 2 21

5+

 √ 112 2 − 127 ;

K [7]; √ 1/2 √ % 1/4 &2 2 2 + 1 + 217/8 2+1 6

64

  b

1 4

;

A.3 Values of K [N ] for all integer N from 1 to 100

K [65] =

2

√ 46/65

65

341

(     √  1/4 √  2 1 + 65 −64 + 7 + 65

1040 V [65]1/8 ⎡                 ⎤1/4 1 9 29 33 37 49 57 61 b 260 b 260 b 260 b 260 b 260 b 260 b 260 b 260 ⎦ , ⎣ × % 7 &  47   51   63  b 260 b 260 b 260 b 260   √ 1/2 √ ; W [65] = 291 + 36 65 + 8 5 524 + 65 65 √   1/8 R 25/88 × 35/8 11 K [66] = 264 T [66] ⎡             ⎤1/2 1 17 9 41 65 b 264 b 88 b 264 b 19 b 88 88 b 264 ⎦ ,   % &     ×⎣ 5 7 29 31 b 264 b 88 b 264 b 88 √ √ √  √ R = 27 − 5 6 − 3 22 − 2 3− 2  √  √ √ 1/2 √ × −29 − 2 6 + 11 11 3 − 6 2 ,  √   √ 3 P[66] = u − u 2 − 1, u = 3 2 3 + 11 '  1/2 √ √ √ × 15 11 − 22 3 + 4 −26 + 6 33 . Note the b’s that appear above with 88 in their denominator are precisely those that appear in K [22], leading to the following remarkable result:  3R 1/4 1 √ √ √ T [66] (33 2 + 7 11 + 5 22)1/2       ⎡ ⎤ 17 41 65 b 264 b 264 b 264 ⎦.     × ⎣ 29 31 b 264 b 264

25/22 K [66]2 = K [22] 24



√    √ 1/3 √ 1/3  67 + 710 − 18 201 K [67] = 8 + 710 − 18 201 804 ⎡                   ⎤1/3 1 4 6 9 14 15 16 17 b 67 b 67 b 67 b 10 b 67 67 b 67 b 67 b 67 b 67 ⎦       % &         ×⎣ 3 5 7 8 12 13 2 b 67 b 67 b 67 b 67 b 11 b b b 67 67 67 67 ⎡                   ⎤1/3 21 22 23 24 25 26 29 33 b 19 67 b 67 b 67 b 67 b 67 b 67 b 67 b 67 b 67 ⎦ ; ⎣               × 20 27 28 30 31 32 b b b b b b b 18 67 67 67 67 67 67 67

342

Appendix

K [68] = K [69] =

 1 + k17 K [17]; 2

√  1/8 R 2173/276 × 35/16 23 552 V [69] ⎡     % &           ⎤1/4 1 5 7 13 25 35 49 65 b 276 b 276 b 276 b 276 b 276 b 276 b 276 b 276 ⎦ , ⎣         × 1 29 37 61 b 276 b 276 b 276 b 92 '  √ √  √ R = 22 − 9 3 + 12 − 8 3 5 + 4 3, 1 V [69]   ' √ √ √ = 2(2 + 3)7 36002 + 27401 3 + 288 138(302 + 177 3) ;

V [69] +

(  √ √ √ √  √ 263/140 × 51/4 × 71/2 K [70] = 35 11 + 5 2 + 6 5 + 6 7 + 3 10 + 3 70 19600 ⎡        ⎤ 1 9 19 59 b 280 b 280 b 280 b 280 √ ⎦ , Q[70] = 3(7 + 2 10); ⎣       × 29 31 b 280 b 18 b 280 √  2/7 71 2 a (a + 1) K [71] = 142 ⎡                 ⎤1/7 1 2 3 4 5 6 8 9 b 71 b 71 b 71 b 71 b 71 b 71 b 71 b 71 ⎢ ⎥       ⎢ ⎥ 12 15 ⎢ ⎥ b b × b 10 71 71 71 ⎢ ⎥             ×⎢ ⎥ % & ⎢ ⎥ 7 11 13 14 17 21 22 b 71 b 71 b 71 b 71 b 71 b 71 b 71 ⎢ ⎥ ⎣ ⎦ ⎡               ⎤1/7 18 19 20 24 25 27 b 16 71 b 71 b 71 b 71 b 71 b 71 b 71 ⎥ ⎢       ⎢ ⎥ 30 32 ⎢ ⎥ × b 29 71 b 71 b 71 ⎢ ⎥ ×⎢              ⎥ , ⎢ b 23 b 26 b 28 b 31 b 33 b 34 b 35 ⎥ ⎢ 71 71 71 71 71 71 71 ⎥ ⎣ ⎦

where a is the only real root of the equation a 7 − 2a 6 − a 5 + a 4 + a 3 + a 2 − a − 1 = 0,

  z = exp − 27 iπ ,

A.3 Values of K [N ] for all integer N from 1 to 100

343

   √ s(n) = Re − 13829 + 2849 −71 z n      √ √ − 43152 + 3752 −71 z 2n − 63690 + 1778 −71 z 3n ; 1, a= 2 + s(1)1/7 + s(2)1/7 − [−s(3)]1/7 + s(4)1/7 + s(5)1/7 + s(6)1/7 , 7 √ √ √ √ 1 + ( 3 + 2)3/2 (14 2 − 10 − 4 6)1/4 K [72] = K [18]; 2   1/8 R 265/73 × 737/16 K [73] = 584 V [73] ⎡ ⎤1/2               1 9 25 37 41 49 57 b 292 b 292 b 292 b 292 b 292 b 292 b 292 ⎢ ⎥       ⎢ ⎥ 61 65 69 ⎢ ⎥ b 292 b 292 × b 292 ⎢ ⎥ × ⎢                ⎥ , ⎢ b 3 b 19 b 23 b 27 b 35 b 55 b 67 b 71 ⎥ ⎢ 292 292 292 292 292 292 292 292 ⎥ ⎣ ⎦ √ 1921 + 225 73 W [73] = ; 2 √ 2227/296 × 379/20 (r1 / 37 + r2 )1/20 K [74] = 592 T [74]1/8 ⎡             ⎤1/5 1 3 9 11 25 27 b 296 b 296 b 296 b 296 b 296 b 296 ⎦     × ⎣ % 7 & 21 b 18 b 296 b 296 ⎡             ⎤1/5 33 41 49 65 67 73 b 296 b 296 b 296 b 296 b 296 b 296 ⎣ ⎦ ,         × 47 53 63 71 b 296 b 296 b 296 b 296 √ R = 125 73 − 1068,

11a 4 + 18a 3 + 43a 2 + 56a + 52 , 2 r1 = 29369a 4 − 24094a 3 + 64167a 2 − 108496a + 37987, Q[74] =

r2 = −204a 4 + 7954a 3 − 10850a 2 + 13232a − 4654, 1, a = − 1 + s(1)1/5 + s(2)1/5 − [−s(3)]1/5 − [−s(4)]1/5 , 5       √ √ s(n) = Re −2 + 100 −74 exp − 25 πin + 948 − 10 −74   × exp − 45 πin ;  √ 1  K [75] = 5 + 2 × 101/3 + (5 + 3 5)10−1/3 K [3]; 15

344

Appendix

K [76] =

 1 + k19 K [19]; 2

   √ √ 1/4 √ √ 11 − 5 13 + 4 11 + 143 + 44 11 2 + 2 11 K [77] = 1/8 616 V [77] ⎡               ⎤1/4 1 3 9 13 27 37 39 b 308 b 308 b 308 b 308 b 308 b 308 b 308 ⎢ ⎥       ⎢ ⎥ 47 53 73 ⎢ ⎥ b 308 b 308 × b 308 ⎢ ⎥             ×⎢ ⎥ , ⎢ ⎥ 5 1 15 23 67 69 b 308 b 44 b 308 b 308 b 308 b 308 ⎢ ⎥ ⎣ ⎦ 2327/616

× 77/16

 √ √ 1229 + 368 11 + 35 2441 + 736 11 ; W [77] = 2  √ √ √ √ √ 1/2 291/156 × 33/4 × 131/4 10 + 15 2 + 12 3 + 17 6 + 3 13 + 4 26 K [78] = 1872 ⎡        ⎤ 1 25 35 49 b 312 b 312 b 312 b 312 ⎦,       ×⎣ 1 23 61 b 312 b 312 b 24  √ 6 √ 3 − 13 P[78] = ( 26 − 5)2 ; 2 √  2/5 79 a3 K [79] = 158 (a − 1) ⎡                 ⎤1/5 1 2 4 5 8 9 11 b 79 b 79 b 79 b 79 b 79 b 79 b 10 79 b 79 ⎥ ⎢       ⎢ ⎥ 16 18 ⎢ ⎥ × b 13 79 b 79 b 79 ⎢ ⎥ ×⎢     % &           ⎥ ⎢ b 3 b 6 b 7 b 12 b 14 b 15 b 17 b 24 ⎥ ⎢ 79 79 79 79 79 79 79 79 ⎥ ⎣ ⎦ ⎡

⎤1/5                b 20 b 21 b 22 b 23 b 25 b 26 b 31 79 79 79 79 79 79 79 ⎢ ⎥       ⎢ ⎥ 36 38 ⎢ ⎥ b b × b 32 79 79 79 ⎢ ⎥ × ⎢                  ⎥ , ⎢ b 27 b 28 b 29 b 30 b 33 b 34 b 35 b 37 b 39 ⎥ ⎢ 79 79 79 79 79 79 79 79 79 ⎥ ⎣ ⎦ 

b

19 79

where a is the only real root of the equation a 5 − 3a 4 + 2a 3 − a 2 + a − 1 = 0,

A.3 Values of K [N ] for all integer N from 1 to 100

345

1, 3 + s(1)1/5 − [−s(2)]1/5 + s(3)1/5 + s(4)1/5 , 5   √ s(n) = Re −(331 + 15 −79) exp(− 25 πin) − 616 exp(− 45 πin) ;

a=

K [80] =

 1 + k20 K [20]; 2    √ 1/3 2 1 + 2(1 + 3)

K [81] = K [82] =

36

 

;   √ 1/2 √ −9 + 41 + −6 + 2 41 b

1 4

2183/328 × 413/8 656 T [82]1/8 ⎡               ⎤1/2 1 9 25 33 43 49 51 b 328 b 328 b 328 b 328 b 328 b 328 b 328 ⎢ ⎥         ⎢ ⎥ 57 59 73 81 ⎢ ⎥ b 328 b 328 b 328 × b 328 ⎢ ⎥             ×⎢ ⎥ , ⎢ ⎥ 5 21 23 31 37 39 b b b b b b ⎢ ⎥ 328 328 328 328 328 328     % & ⎣ ⎦ 45 61 77 × b 328 b 328 b 328

√ 3 (19 + 3 41); 2 √ 1/3 √ 188/9 × 83  √ (5 249 + 9)1/3 − (5 249 − 9)1/3 R2 K [83] = 5976 ⎡       % &         ⎤1/9 1 3 4 7 9 11 12 b 83 b 83 b 83 b 83 b 10 b 83 83 b 83 b 83 ⎥ ⎢           ⎢ ⎥ 17 21 23 25 ⎢ ⎥ × b 16 83 b 83 b 83 b 83 b 83 ⎢ ⎥ × ⎢                ⎥ ⎢ b 2 b 5 b 6 b 8 b 13 b 14 b 15 b 18 ⎥ ⎢ 83 83 83 83 83 83 83 83 ⎥ ⎣ ⎦ Q[82] =

⎡                 ⎤1/9 27 28 29 30 31 33 36 b 26 83 b 83 b 83 b 83 b 83 b 83 b 83 b 83 ⎥ ⎢         ⎢ ⎥ 38 40 41 ⎢ ⎥ × b 37 83 b 83 b 83 b 83 ⎢ ⎥ ×⎢                ⎥ , ⎢ b 19 b 20 b 22 b 24 b 32 b 34 b 35 b 39 ⎥ ⎢ 83 83 83 83 83 83 83 83 ⎥ ⎣ ⎦ 2 + (β + γ )1/3 + (β − γ )1/3 , 3     √ 1/3 √ 1/3 1 + 3 × 22/3 79 + 3 249 β= 29 + 3 × 22/3 79 − 3 249 , 27     √ 1/3 1/2 √ 1/3 1 + 4 1432 + 3 249 ; 49 + 4 1432 − 3 249 γ = 9 R=

346

Appendix

K [84] =

 1 + k21 K [21]; 2

√ √ √ √ 1/4 213/170 × 85  954 + 392 5 + 216 17 + 105 85 340 ⎡                   ⎤1/2 1 9 21 37 49 57 69 73 81 b 340 b 340 b 340 b 340 b 340 b 340 b 340 b 340 b 340 ⎦ ,   % &           ×⎣ 3 7 19 23 27 59 63 b 340 b 340 b 340 b 340 b 340 b 340 b 340 √ 3901 + 945 17 W [85] = ; 2

K [85] =

√ 21341/1720 × 439/20 (r1 / 2 + r2 )1/20 K [86] = 592 T [86]1/8 ⎡             ⎤1/5 1 3 9 17 19 25 b 344 b 344 b 344 b 344 b 344 b 344 ⎣ ⎦       × % 7 & 13 21 39 b 344 b 344 b 344 b 344 ⎡               ⎤1/5 27 41 49 51 57 75 81 b 344 b 344 b 344 b 344 b 344 b 344 b 344 ⎦ ,         ×⎣ 53 55 63 71 b 344 b 344 b 344 b 344 3a 4 + 10a 3 + 13a 2 + 4a − 2 , 2 r1 = −2192a 4 − 29128a 3 − 59308a 2 + 94876a + 196322, Q[86] =

r2 = −1563a 4 − 21058a 3 − 41223a 2 + 67132a + 139169, 1, −1 + s(1)1/5 + s(2)1/5 − [−s(3)]1/5 + s(4)1/5 , a= 5     √ s(n) = Re −1158 + 20 −86 exp − 25 πin     √ − 1288 + 50 −86 exp − 45 πin ;   √ 1/3  √ 1/3 1/3 √  √ 1/6 + 23+2 29 87 2 + 23−2 29 32(5 + 29) K [87] = 348 ⎡                   ⎤1/3 1 2 8 13 17 22 25 26 b 87 b 87 b 87 b 87 b 14 87 b 87 b 87 b 87 b 87 ⎦ ; ⎣             × 1 5 23 38 40 b 87 b 19 b b b b 29 87 87 87 87 K [88] =

 1 + k22 K [22]; 2

A.3 Values of K [N ] for all integer N from 1 to 100

347

2167/267 × 8923/48 R 1/24 712 V [89]1/8 ⎡               ⎤1/6 1 5 9 17 21 25 45 b 356 b 356 b 356 b 356 b 356 b 356 b 356 ⎣ ⎦         × 11 39 47 55 b 356 b 356 b 356 b 356 ⎡               ⎤1/6 49 53 57 69 73 81 85 b 356 b 356 b 356 b 356 b 356 b 356 b 356 ⎦ ,         ×⎣ 67 71 79 87 b 356 b 356 b 356 b 356    √ 15649 1 (u − v) R= −15580 + 7583 89 + 4 1591 − √ 3 89    81 + 108 √ − 23 (u 2 + v 2 ) , 89 √ √ u = (−91 + 6 267)1/3 , v = (91 + 6 267)1/3 , √ √ 1  267(4879 + 519 89) − 4(48683 + 5149 89)(u − v) W [89] = 1602  √ + 22(2314 + 245 89)(u 2 + v 2 ) ;

K [89] =

K [90] =

√ √ √ √ 6( 2 − 1)( 5 − 2 + 1) 6  √ √ √ √ √ √ √ 1/2  −45 + 33 2 − 54 3 − 21 5 + 39 6 + 15 10 − 24 15 + 17 30 + 3 × K [10];

(   √ 13 − 3 75/6 26 K [91] =

R2 728 ⎡                 ⎤1/3 5 1 9 22 23 24 25 b 91 b 13 b 91 b 20 91 b 91 b 91 b 91 b 91 ⎥ ⎢       ⎢ ⎥ 33 36 ⎢ ⎥ × b 31 91 b 91 b 91 ⎢ ⎥ ×⎢                ⎥ , ⎢ b 2 b 3 b 8 b 15 b 17 b 27 b 32 b 40 ⎥ ⎢ 91 91 91 91 91 91 91 91 ⎥ ⎣ ⎦

   √ 1/3  √ 1/3 √ √ 1 + 44 + 9 13 + 3 21 2 + 44 + 9 13 − 3 21 ; R= 3    √ √ 1/3 √ K [92] = 24 + 61/2 48 − 4 23 + 5 × 22/3 −21 3 + 17 23    √ √ 1/3 1/2 K [23] 2/3 21 3 + 17 23 +5×2 ; 48

348

Appendix √ √ √ 1/4 224/31 × 933/8  2532 + 1170 3 + 455 31 + 210 93 744 ⎡               % &   ⎤1/2 1 17 25 29 49 53 65 77 89 b 372 b 372 b 372 b 372 b 372 b 372 b 372 b 372 b 372 ⎦ ; × ⎣ % 7 &  11   19   23   67   83   1  b 372 b 372 b 372 b 372 b 372 b 372 b 4 √ √ √ V [93] = (1520 − 273 31)2 (45 3 − 14 31)2 ;

K [93] =

 1/8 477/16 R 376 2T [94] ⎡               ⎤1/4 1 9 11 17 19 25 35 b 376 b 376 b 376 b 376 b 376 b 376 b 376 ⎦           × ⎣ 15 21 23 31 37 b 376 b 376 b 376 b 376 b 376 ⎡               ⎤1/4 43 49 65 67 81 89 91 b 376 b 376 b 376 b 376 b 376 b 376 b 376 ⎦ ,           ×⎣ 39 53 61 87 b 376 b 18 b 376 b 376 b 376 √ √ R = −26 − 32 2 + (23 − 5 2)u   '  √ √ √ 1 + 2 + 2(7 − 4 2 + u) 17 + 5 2 + 13(1 − 2)u , 2 ' '   √ √ √ 1 39 + 30 2 + 3 345 + 244 2 ; u = 9 + 8 2, Q[94] = 2 √ 19/24 5/16 13/24 2 ×5 ( 5 − 1) × 193/8 5/6 1/3 R S K [95] = 3040            ⎤1/2  ⎡ 1 6 16 26 36 b 95 b 11 b 95 95 b 95 b 95 b 95 ⎦ ,         ×⎣ 1 29 34 b b b b 14 95 5 95 95 ' √ √ √ R = 5 + ( 5 + 2) 2 5 − 1 1/2  ' √ √ + ( 5 + 2)1/2 6 + 2 5(2 5 − 1) ,  ' √ √ √ S = −(3 + 5) 2 5 − 1 + 5(2 5 − 1) '   √ 1/2 √ 1/2 ; −2 13 − 10 5 + 91 + 122 5 K [94] =

K [96] = (R −



R 2 − R)K [24],

R = (2 +

√ √ 2 √ 3) ( 3 + 2)2 ;

A.3 Values of K [N ] for all integer N from 1 to 100

349

 1/8 R 2187/194 × 977/16 776 V [97] ⎡             ⎤1/2 1 9 25 33 49 53 b 388 b 388 b 388 b 388 b 388 b 388 ⎦           × ⎣ 3 11 27 31 45 b 388 b 388 b 388 b 388 b 388 ⎡               ⎤1/2 61 65 73 81 85 89 93 b 388 b 388 b 388 b 388 b 388 b 388 b 388 ⎦ , ⎣             × 43 47 75 79 91 95 b 388 b 388 b 388 b 388 b 388 b 388 √ √ 7537 + 765 97 R = 569 97 − 5604, W [97] = ; 2 ' √ √ K [98] = M7 (2)K [2], Q[98] = 23 + 6 14 + 2 2(119 + 32 14).

K [97] =

One of the simplest equations for the seventh-order multiplier seems to be 7M7 (μ) −

1 = 6 − 16t + 12t 2 − 8t 3 , M7 (μ)

t 4 = kμ k49μ ,

and alternatives originating with Ramanujan are studied in Section 9.5 of the Borweins’ Pi and the AGM. √ √ [9 + 2(21 33 − 27)1/3 + 2(54 + 6 33)1/3 ]1/2 K [99] = K [11]; 27 √ √ 4 + 2 5 + 2(3 + 2 × 51/4 )  1  K [100] = b 4 . 80

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Index

β, see beta function (k, l)± , see Hurwitz zeta function (k, l : s), see Hurwitz zeta function (m|n), see Kronecker symbol (s)n , see Pochhammer symbol p Fq , see hypergeometric function G, see Catalan’s constant Iν (x), see modified Bessel function Jν (x), see Bessel function (k), 204 K (k), K [N ], see elliptic integral K ν (x), see modified Bessel function L-series, see Dirichlet L-series L k (s), see Dirichlet L-series Ms [ f (t)], see Mellin transform q-series, 33, 65, 114, 198, 200, 206, 216, 217 Q(a, b, c; s), see two-dimensional lattice sum Q 0 , Q 1 , Q 2 , Q 3 , see Jacobi’s products Q i S(a, b, c; s), see two-dimensional lattice sum S( p, r, j; s), see displaced lattice sum T ( p 2 , 0, −r 2 ; s), see quadratic form, indefinite W B , W F , W S , see Watson integrals α(s), see Dirichlet eta function β(s), see Dirichlet beta function γ , see Euler–Mascheroni constant (z), see gamma function Z | hg |, see Epstein zeta function ζ (s), see Riemann zeta function ζ (s, q), see Hurwitz zeta function η(τ ), see Dedekind eta function η(s), see Dirichlet eta function θ ( j, p), 164, 165 θ (k, l), 202 θ2 (q), θ3 (q), θ4 (q), see theta functions  , 12, 20, 28, 129, 130, 150, 179 φ(k, l), 202 φ(q), see Ramanujan notation χ (k, l), 202 χk (n), see Dirichlet L-series

ψ(q), see Ramanujan notation ?

=, 243 Abel, Niels, 304, 313 Abel summable expression, 109, 117 Abelian group, 58, 218 absolute convergence, xiii, 3, 10, 87, 91, 101, 110, 150, 269 alternating series, see test, alternating series, 59, 97, 100, 112, 184, 222, 289 analytic continuation, 3, 39, 101, 103, 104, 110, 111, 113, 120, 178, 219, 222, 228–231, 236, 238, 257, 259, 290 Andrews, George, 116 angular-dependent sums, 138, 144, 148 anisotropic lattice, 306, 308 Appell, P., xvii, 1, 2, 9, 30, 301–303, 308, 312 asymptotics, 29, 43, 53, 149, 179, 297, 303 Bailey, D. H., 314 Bailey, W. N., 301, 302 Benson, G. C., xiii, 27, 29, 43 Berndt, B. C., 170, 171, 186 Bernoulli, Jacob Bernoulli numbers, 181 Bernoulli polynomials, 158, 290 Bessel function, 17, 68, 135, 140, 253, 269, 270, 302, 316 addition theorem, 141 modified Bessel function, 6, 27, 43, 44, 68, 74, 132–135, 146, 149, 151, 222, 243, 314, 316 recurrence, 137, 143, 316 table for 2D sums, 44 beta function, 74, 75, 120, 180, 317 binomial theorem, 68 Bloch factor, 140 body-centred cubic (BCC) lattice, 64, 73, 228, 234, 236, 295 Born, Max, xvii, 1, 4, 14, 15, 20, 21, 42, 228

Index Born’s Grundpotential, 15, 20 Borwein, D., xi, 229, 232, 259, 265, 285, 289, 290 Borwein, J. M., xi, 75, 192, 197, 229, 232, 259, 265, 285, 289, 290, 305, 349 Borwein, P. B., xi, 192, 197, 229, 349 Bravais lattice, 9, 10, 15, 20, 21, 23, 34, 235 Broadhurst, D., 76, 314

cardinal point, 146–148 Catalan, Eugène, 159 Catalan’s constant, 72, 138, 178, 239, 292 Cauchy, Augustin-Louis, 106, 110, 114 Cauchy principal value, 142 Cauchy residue theorem, 63, 290 central beta function, see beta function Chaba, A. N., 30, 43, 47, 50, 52 character, see Dirichlet L-series Chowla, S., 66, 177 class number, 55, 58, 76, 159, 221 Clausius, Rudolf, 126 closed form, 28, 30, 48, 63, 66, 73, 78, 116, 122, 128, 143, 146, 157, 164, 174, 175, 177, 179, 181, 200, 202, 215, 216, 220, 229, 237, 242, 295, 299, 302, 306, 311, 312 computer algebra system, see also Maple, Mathematica, experimental mathematics, 198, 290, 307, 319 conditional convergence, xi, 10, 26, 39, 64, 118, 127, 137 conical lattice sums, 116 conjectures, 59, 65, 73, 77, 163, 194, 217, 219, 243, 314 contour integration, 63, 114, 146, 147, 217 Coulomb, Charles-Augustin de, 14, 30, 40, 215, 217, 229, 241 Crandall, R. E., xiii, 116, 118, 253 critical line, 144, 148, 150, 152, 153, 179 crystal structures, see also BCC, FCC, SC, etc. CaF2 , 9, 40, 235, 236 CsCl, 29, 40, 117 NaCl, xi, xvii, 12, 15, 16, 40, 72, 88, 93, 104, 115, 117, 213, 238, 240, 247, 260 ZnS, 16, 117 cylinder, 126–128

de Wette, F. W., 26, 52, 64 Debye, Peter, 117 Dedekind, Richard Dedekind eta function, 78, 204, 205, 217 Delves, R. T., 306, 309–311 diamond lattice, 234, 313 differential equation, 303, 307, 312, 319 differentiation under the integral sign, 143 dipole sums, 23, 26, 52, 126, 127, 231 Dirac, Paul Dirac delta function, 114 Dirichlet, Johann, 55

365

Dirichlet L-series, 54, 56–58, 78, 117, 148, 150, 157, 166, 168, 178, 194, 195, 206, 213, 215, 220, 242, 250, 260, 290 table for 2D solutions, 60 table for 3D sums in terms of a single L-series, 213 Dirichlet beta function, 28, 33, 39, 43, 47, 52, 66, 114, 115, 159, 183 Dirichlet character, see Dirichlet L-series character table, 330 primitive character, 158, 187, 188 Dirichlet eta function, 33, 43, 116, 257 displaced lattice sum, 146, 164, 178 distributive lattice sum, 140, 142 divergence, 20, 39, 88, 94, 95, 100, 106, 109, 117, 148, 178, 227, 229 Domb, C., 297, 303 Eisenstein, G. Eisenstein series, 178, 179, 197 electron sums, 226 ellipse, ellipsoid, 17, 250, 251, 259, 266 elliptic curve, xi, xiv, 217, 218 elliptic integral, 52, 66–70, 75, 77, 122, 171, 180, 218, 241, 295, 303, 304, 316, 317 K[N], 73 list of values for N from 1 to 100, 331 Emersleben, O., 15, 16, 27, 42, 54, 94 Epstein, P., 2–4, 10, 11, 17, 27 Epstein zeta function, 2, 15 Euler, Leonard, xiv, 204, 207, 211, 216, 217, 297 Euler totient function, 157 Euler–Mascheroni constant, 28, 52, 195 Evjen, H., 20, 88 Ewald, P., 9–11, 14–16, 20, 21, 23, 25, 30, 42, 44, 65, 88, 235 Ewald method, 9, 22, 233 experimental mathematics, xviii, 59, 76, 78, 118, 150, 163, 168, 175, 179, 194, 197, 219, 220, 243, 314, 319 face-centred cubic (FCC) lattice, 19, 21, 40, 70, 74, 228, 234, 295, 308, 312, 315 Faraday, Michael, 125 Ferguson, H., xi Fermat, Pierre de, 200, 217 Feynman, Richard, 118, 314 five-dimensional sums, 220 fluorite, see crystal structures, CaF2 Foldy, L. L., 234, 235 Forrester, P. J., 245 four-dimensional sums, 34, 120, 195, 216, 223, 237 Fourier, Joseph, 14, 30 Fourier series, 78, 134, 135 Fourier transform, 2, 25, 311, 316 fast, 118 gamma function, 52, 66, 68, 74, 75, 78, 102, 117, 128, 180, 241, 290, 295, 296, 304

366

Index

incomplete gamma function, 22, 29 Gauss, Carl, 68, 150, 179, 208, 307, 320 Gauss circle problem, 90, 113, 261 Gaussian curvature, xi, xiii Gaussian quadrature, 181 generating function, 67, 181, 319 Glasser, M. L., xviii, 30, 33, 47, 50, 59, 65–67, 72, 77, 131, 134, 258, 303, 304 Green, George Green’s functions, 53, 70, 73, 130, 131, 133, 135, 137, 141, 164, 298, 310, 315 Guttmann, A. J., 74, 312 Hankel, Hermann, 43, 140, 141 Hardy, G. H., 28, 90, 113, 116, 146, 183, 250, 295 Helmholtz equation, 140 hexagonal close-packed (HCP) lattice, 233, 234 hexagonal lattice, 56, 72, 109, 229, 232, 233, 259 Hobson’s integral, 27, 132, 149, 221, 222 Hund, F., 15, 16, 19, 21, 27, 34–36, 39, 42, 233 Hund potentials, table, 37 Hurwitz, Adolf, 4 Hurwitz zeta function, 159, 162, 176, 290 hyperbolic function, 29, 44, 46, 47, 49, 63, 64, 128, 217, 218, 223, 232, 240 hypercube, 230, 231, 260, 313, 315 hypergeometric function, 68, 70, 73, 77, 136, 178, 216, 218, 220, 296, 303, 316, 319 Appell hypergeometric function, 301 transformation, 200, 306, 307, 311 inequality, 91, 96 triangle inequality, 112 integer relation program, see PSLQ integration by parts, 18, 77, 272, 291, 316 invariant cubic lattice complexes (ICLCs), 32, 36, 39, 42 iPhone, 118 Iwata, G., 299, 301–303, 308, 310, 313 Jacobi, Carl, 3, 22, 27, 31, 32, 56, 114, 130, 132, 134, 158, 160, 171, 187, 203–205, 209, 210, 212, 216, 217, 296, 311 Jacobi symbol, see Kronecker symbol Jacobi theta function, see theta function Jacobi’s products Q i , 31, 204 triple product, 209 jellium, 137, 226 Joyce, G. S., 67, 68, 70, 73, 299, 303, 305, 310, 313 Kepler, Johannes, xiv Klein, Felix, 77 Kronecker, Leopold, 4, 66, 138, 181 Kronecker’s Grenzformel, 66 Kronecker symbol, 56, 76, 158, 187, 197, 257 Lambert, Johann Heinrich

Lambert series, 160, 186, 189, 192, 197, 200, 209, 210 Landé, Alfred, xvii, 9 Landau, E., 17, 18, 251, 252, 267, 269 Laplace, Pierre-Simon Laplace equation, 2, 6, 7, 126, 133, 135 Laplace transform, 26, 113, 216 Laplacian, 131, 298 lattice constants, xvii, 20, 72 lattice sum definition of, 2, 248 table of exponential and hyperbolic forms, 49 table of numerical values, 51 table of sums, involving indefinite quadratic forms, 176 Legendre, Adrien-Marie, 56, 190, 304 Legendre symbol, see Kronecker symbol Liouville, Joseph, 153, 223 Lord Rayleigh, 125–127, 130, 134, 144 Lorentz H. A., 125 Lorenz, L., 20, 26, 28, 33, 113, 125, 146, 175, 183 Macdonald functions, see modified Bessel functions Madelung, E., xvii, 4, 7, 8, 12, 20, 28–30, 43, 117, 118, 238 Madelung constant, xi, xvii, 12, 15, 16, 36, 72, 87, 88, 93–95, 100, 101, 104, 109, 111, 114, 116, 118, 184, 217, 237, 247, 259, 260, 290 table of Madelung constants for ionic crystals, 41 magnetism, 9, 68, 295 Mahler, K. Mahler measure, 73, 78, 219, 314 Maple, 125, 130, 179, 199, 319 Maradudin, A. A., 29, 226, 229, 295 Mathematica, 76, 125, 130, 169 McPhedran, R. C., 138, 140, 148, 164, 168, 170 mean value theorem, 91 Mellin, H. Mellin transform, 30, 32, 36, 65, 78, 101, 102, 110, 132, 149, 151, 152, 160, 164, 184, 186, 189, 190, 192–194, 206, 207, 209–211, 216, 221, 223, 239, 285, 290 of θ -function (MTθ F), 30 table, 209 method of expanding geometric shapes, xiii, 89, 94, 95, 106, 109, 249, 259, 261, 263 modular equation, 186, 193–195, 197, 199, 211, 305, 325–329 modular equation order, 3, 110, 188, 190–193, 211 tables for order 3, 325–327 tables for order 5, 328 tables for order 7, 329 modular form, 55, 74, 78, 197, 217, 219, 273 modular group, 197 monotonicity, 91, 93, 95, 96, 98, 268, 289

Index Movchan, A. B., 138 MTθ F, 30 multipole, 16, 25, 26, 126–128 Naor potentials, table, 37 Newton, Isaac, 183 Newton-John, Olivia, xvii Nicorovici, N. A., 138 Nijboer, B. R. A., 25, 26, 30, 52 number theory, xiv, 18, 54, 56, 89 numerical methods, xviii, 4, 16, 20, 29, 30, 39, 42, 44, 51, 54, 65, 72, 128, 142, 146, 148–151, 175, 178, 179, 181, 219, 233, 241, 247, 297, 313, 315 optical properties, 26, 125 orthogonal polynomials, 181 orthonormalization, 183 orthorhombic lattice, 50, 310 Parseval’s theorem, 25 partial fractions, 178, 290 Pathria, R. K., 30, 43, 47, 50, 52 perovskite, 72, 235 phase-modulated sums, 131, 164 Pochhammer, L. A. Pochhammer symbol, 73, 139, 178, 291, 311 Poisson, Siméon Denis, 182 Poisson equation, 6, 8, 237, 315 Poisson summation formula or transform, 7, 29, 30, 43, 44, 45, 50, 132, 140, 142, 149, 164, 190–192, 205, 207, 211, 215, 221, 227 table, 209 pole, simple, 4, 39, 45, 63, 142, 145, 147 power series, 128, 176, 181, 302 prime number, 54, 56, 118, 153, 157, 160, 161, 187, 218 probability, 151, 296 PSLQ algorithm, xi, 179, 199, 243, 320 quadratic form, 2, 18, 55, 56, 58, 113, 159, 166, 175, 177, 186, 193, 216, 236, 248–250, 252, 254, 256, 258, 264, 283 discriminant, 55, 56, 58, 59, 76, 116, 159, 175, 177, 258 genus, 58, 59, 177, 194, 258 indefinite, 175 quadratic residue, see Kronecker symbol Ramanujan, S., 73, 78, 116, 171, 178, 186, 189, 193, 199, 200, 208, 211, 212, 304, 349 random walk, 67, 68, 78, 152, 294, 296, 302, 306 reciprocal lattice, 3, 9, 10, 13, 23, 42, 131, 134, 135, 139–141, 144 Brillouin zone, 67, 146–148 rectangular lattice, 7, 131, 151 recurrence, 23, 129, 137–139, 151, 180, 182, 291, 303, 307, 319 recursion, see recurrence

367

restricted sums, 112 rhombohedral lattice, 29 Riemann, Bernhard, xiii, 3, 9 generalized Riemann hypothesis, 148 Riemann hypothesis, 116, 147, 148, 151, 153, 179 Riemann zeta function, 2, 45, 54, 110, 125, 152, 159, 180, 183, 188, 220, 223, 228, 242, 290 Robertson, M. M., 34, 54, 59, 157, 158, 161 rock salt, xvii, 4, 8, 28, 217, see also crystal structures Rogers, Mathew, 74, 78, 178, 218, 219, 315 Sakamoto, Y., 39, 42 Selberg, A., 66, 75 series, slowly convergent, 7, 10, 25, 50, 118 sign transform, 186, 190, 192, 204, 207, 211, 212 simple cubic (SC) lattice, 14, 15, 21, 22, 40, 67, 70, 74, 126, 226, 228, 234, 235, 237, 295, 302, 306, 310, 313, 315 singular value, 68, 70, 75, 304, 317, 318 k210 , 116 table of singular values of K [N ], 331 spanning tree, 72, 314 spinel, 72, 235 square lattice, 72, 114, 127, 129, 134, 135, 138, 140, 144–146, 164, 232, 316 Stirling, James Stirling numbers, 291 symmetry, mathematical, 5, 15, 21, 26, 28 tables L-series solutions of 2D sums, 60 2D Bessel sums, 44 3D sums in terms of a single Dirichlet L-series, 213 character table, 330 exponential and hyperbolic forms for lattice sums, 49 expressions in Eulerian and Jacobian forms, 208, 210 lattice sums involving indefinite quadratic forms, 176 Madelung constants for ionic crystals, 41 modular equations of order 3, 325–327 modular equations of order 5, 328 modular equations of order 7, 329 Naor and Hund potentials, 37 numerical values of lattice sums, 51 Poisson and Mellin transforms, 209 singular values of K [N ], 331 Taylor series, 130, 131, 145, 151, 182, 198, 199, 295 term-by-term integration, 241, 262, 306, 308 test alternating series, 289 comparison, 101 Weierstrass M-test, 108, 263, 265

368

Index

tetragonal lattice, 50, 306 theta functions, 22, 30, 34, 87, 110, 113, 114, 130, 132, 134, 147, 160, 165, 178, 184, 186, 200, 203, 216, 241 conical, 116 multiplicative relations, 32 table of Eulerian and Jacobian forms, 208, 210 θ1 , 66, 134 θ1 , 31, 39, 66, 134, 212 θ2 , 34, 36, 39, 45, 46, 78, 103, 104, 189, 211, 223 θ3 , 32, 34, 36, 39, 45, 46, 78, 189, 193, 195, 210, 211 θ3 functional equation, see Poisson summation formula θ4 , 32, 34, 39, 45, 46, 78, 101, 112, 189, 192, 211 θ5 , 31, 34, 45, 186, 189, 203 θ6 , 31 three-dimensional sums, 34, 39, 52, 94, 184, 202, 215, 221, 232, 240, 256, 292 Tosi, M. P., 1, 16, 21, 29, 40 tree, see spanning tree

triangular lattice, see hexagonal lattice two-dimensional lattice sum, 55, 57, 59, 60, 89, 105, 109, 160, 177, 178, 184, 193, 256, 292 Vandermonde, A., 183 Wan, J. G., 78, 184 Watson, G. N., 31, 43, 53, 67, 270, 294, 298, 301, 302, 304 Watson integrals, 53, 67–70, 73, 294 wave, 140–142 Weber’s integral, 143 Weierstrass, Karl, 130 Weierstrass elliptic function, 54, 130, 180 Weierstrass M-test, 108, 263, 265 Wigner, E. P., 150, 226, 236 Wiles, Andrew, 217, 219 WIs, see Watson integrals Wu, F. Y., 72 Zeilberger, D., 319 Zucker, I. J., xviii, 32, 33, 45, 47, 50, 52, 54, 59, 64–68, 75, 78, 147, 157, 158, 161, 170, 194, 212, 221, 229, 235, 258, 288, 304, 313 Zudilin, W., 218, 219