The Theory of Zeta-Functions of Root Systems 9789819909094, 9789819909100

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Springer Monographs in Mathematics

Yasushi Komori Kohji Matsumoto Hirofumi Tsumura

The Theory of Zeta-Functions of Root Systems

Springer Monographs in Mathematics Editors-in-Chief Minhyong Kim, School of Mathematics, Korea Institute for Advanced Study Seoul, South Korea, International Centre for Mathematical Sciences, Edinburgh, UK Katrin Wendland, School of Mathematics, Trinity College Dublin, Dublin, Ireland Series Editors Sheldon Axler, Department of Mathematics, San Francisco State University, San Francisco, CA, USA Mark Braverman, Department of Mathematics, Princeton University, Princeton, NJ, USA Maria Chudnovsky, Department of Mathematics, Princeton University, Princeton, NJ, USA Tadahisa Funaki, Department of Mathematics, University of Tokyo, Tokyo, Japan Isabelle Gallagher, Département de Mathématiques et Applications, Ecole Normale Supérieure, Paris, France Sinan Güntürk, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA Claude Le Bris, CERMICS, Ecole des Ponts ParisTech, Marne la Vallée, France Pascal Massart, Département de Mathématiques, Université de Paris-Sud, Orsay, France Alberto A. Pinto, Department of Mathematics, University of Porto, Porto, Portugal Gabriella Pinzari, Department of Mathematics, University of Padova, Padova, Italy Ken Ribet, Department of Mathematics, University of California, Berkeley, CA, USA René Schilling, Institute for Mathematical Stochastics, Technical University Dresden, Dresden, Germany Panagiotis Souganidis, Department of Mathematics, University of Chicago, Chicago, IL, USA Endre Süli, Mathematical Institute, University of Oxford, Oxford, UK Shmuel Weinberger, Department of Mathematics, University of Chicago, Chicago, IL, USA Boris Zilber, Mathematical Institute, University of Oxford, Oxford, UK

This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

Yasushi Komori · Kohji Matsumoto · Hirofumi Tsumura

The Theory of Zeta-Functions of Root Systems

Yasushi Komori Department of Mathematics Rikkyo University Tokyo, Japan Hirofumi Tsumura Department of Mathematical Sciences Tokyo Metropolitan University Tokyo, Japan

Kohji Matsumoto Graduate School of Mathematics Nagoya University Nagoya, Japan Center for General Education Aichi Institute of Technology Toyota, Japan

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-981-99-0909-4 ISBN 978-981-99-0910-0 (eBook) https://doi.org/10.1007/978-981-99-0910-0 Mathematics Subject Classification: Primary 11M32, Secondary 11B68, 11M41, 11M99, 17B10, 17B22, 32D15, 32S22, 52B11 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface The aim of this book is to give the full account of the present stage of the theory of zeta-functions of root systems, which are certain multiple series associated with root systems. The theory of multiple zeta-functions has a long history going back to the eighteenth century, but the modern active research began in the 1990s. In this trend, the central role has been played by Euler–Zagier multi-variable multiple zeta-functions. Special values of them, called multiple zeta values (MZVs for short), have close connections with various fields of mathematics such as arithmetic geometry, topology and so on. Some variants of Euler–Zagier multiple zeta-functions, for instance Mordell–Tornheim multiple zeta-functions, have also been introduced and studied extensively. On the other hand, in the field of quantum gauge theory in mathematical physics, the notion of Witten zeta-functions associated with semisimple Lie algebras was introduced. Witten zeta-functions can be written down as multiple series, and hence those are also multiple zeta-functions, but in one variable. Zeta-functions of root systems give a unified view in the above landscape of the theory of multiple zeta-functions. The history of the theory of zeta-functions of root systems can be traced back to a paper of L. Tornheim in 1950, but the general definition was given by the authors. The original motivation of the authors was to consider the multi-variable version of Witten zeta-functions; first the definition of the B2 case was introduced by the second-named author in 2003, and then the definition of the general case was published in 2007. The research of the authors on this theory has been ongoing for decades. An advantage of introducing the multi-variable version is that we can get more analytic flexibility. Because of this flexibility, we can discuss the recursive structure in the family of Witten zeta-functions, and can achieve the meromorphic continuation of them.

v

vi

PREFACE

Another important point is that the notion of zeta-functions of root systems is not just the multi-variable generalization of Witten zetafunctions, but also, a generalization of multi-variable multiple zetafunctions of Euler–Zagier type and of Mordell–Tornheim type. Therefore the notion of zeta-functions of root systems is unification of various important classes of multiple zeta-functions. Moreover, the family of zeta-functions of root systems has various fascinating structures, coming from the structure of the underlying root systems. Using these structures, we can develop the theory of functional relations of zeta-functions of root systems. From the same viewpoint, it is also possible to apply the theory of zeta-functions of root systems to the study of MZVs, and new methods and results on MZVs have been obtained. Almost all of the contents of the present book is the output from the authors’ papers. The main topics treated in this book are special values of zeta-functions of root systems at positive integer points, the meromorphic continuation, the functional relations, applications to MZVs, some variants and generalizations, and so on. We not only give the general theory, but also write down a lot of explicit examples in the case of zeta-functions of root systems of low ranks. The basic theory of Lie algebras and root systems, necessary to read this book, is gathered in Chapter 2, and further required knowledge is explained here and there in the text. Many theorems in this book are not merely reproduced from the original papers. Some theorems are proved in a way different from the original one. Some theorems are stated in a generalized form, with a generalized proof. Misprints and inaccuracies included in the original papers are corrected. Furthermore, there are some results in this book which have not been published before. The authors would like to thank Springer-Verlag for publishing this book, in particular Mr. Masayuki Nakamura of Springer Tokyo for valuable help with its arrangement. The authors also express their sincere gratitude to Professors Hidekazu Furusho, Shin-ya Kadota, Masanobu Kaneko, Hideki Murahara, Takashi Nakamura, Maki Nakasuji, Hiroyuki Ochiai, Yasuo Ohno, Masataka Ono, Tatsushi Tanaka, Isao Wakabayashi, and anonymous referees for valuable comments and suggestions. December 2022 Yasushi Komori, Kohji Matsumoto and Hirofumi Tsumura

Contents Preface

v

Chapter 1. Introduction 1.1. The Euler double zeta-function 1.2. Multiple zeta-functions of Barnes and of Mellin 1.3. Euler–Zagier multiple zeta-functions 1.4. Mordell–Tornheim multiple zeta-functions 1.5. Witten multiple zeta-functions 1.6. Zeta-functions of root systems

1 1 3 6 10 12 12

Chapter 2. Fundamentals of the theory of Lie algebras and root systems 15 2.1. Lie algebras 15 2.2. Roots 17 2.3. Abstract root systems 19 2.4. Weights 21 Chapter 3. Definitions and examples 3.1. The definition of zeta-functions of root systems 3.2. Explicit forms for zeta-functions of type A 3.3. Explicit forms for zeta-functions of other types

25 25 29 31

Chapter 4. Values at positive even integer points 4.1. The sum S(s, y; ∆) 4.2. The values at even integer points 4.3. A closed expression of F (t, y; ∆) and further examples

39 39 43 50

Chapter 5. Convex polytopes and the rationality 5.1. Convex polytopes 5.2. The rationality

59 59 66

Chapter 6. 6.1. The 6.2. The 6.3. The

71 71 73 76

The recursive structure Mellin–Barnes integral formula recursive structure for Ar recursive structure for Br , Cr and Dr

vii

viii

CONTENTS

6.4. Recursive structures and Dynkin diagrams Chapter 7. The meromorphic continuation 7.1. The meromorphic continuation of zeta-functions of root systems 7.2. Multiple zeta-functions defined by linear forms

84 91 91 97

Chapter 8. Functional relations (I) 113 8.1. A method to evaluate the Riemann zeta-function 114 8.2. Functional relations for the zeta-function of A2 120 8.3. Functional relations for the zeta-function of C2 137 8.4. An application of Nakamura’s method to the zeta-function of G2 149 Chapter 9. Functional relations (II) 9.1. A sketch in the case of A2 9.2. Lemmas on root systems and Weyl groups 9.3. Automorphisms on Dynkin diagrams 9.4. General functional relations 9.5. An extension 9.6. The action of Aut(∆) 9.7. Explicit generating functions (#I = (r − 1) case) 9.8. Explicit functional relations (#I = (r − 1) case) 9.9. Explicit functional relations (#I = 1 case)

159 159 161 164 166 171 177 183 197 209

Chapter 10. Poincar´e polynomials and values at integer points 10.1. Poincar´e polynomials 10.2. The non-vanishing of the sum of coefficients

213 213 218

Chapter 11.1. 11.2. 11.3. 11.4.

11. The case of the exceptional algebra G2 A criterion of reduction Application of Theorem 11.1 to the case G2 Some lemmas A functional relation corresponding to I = {2}

223 223 226 227 233

Chapter 12.1. 12.2. 12.3. 12.4. 12.5.

12. Applications to multiple zeta values (I) The case of type A2 The case of type Cr The case of type Br Restricted sum formulas Parity results

241 241 244 251 258 261

Chapter 13. Applications to multiple zeta values (II)

263

CONTENTS

13.1. 13.2. 13.3. 13.4. 13.5.

ix

Hoffman’s algebra Extension of Hoffman’s setup Double and triple zeta values Proofs of results in Section 13.2 Functional relations including double shuffle relations

263 266 268 270 275

Chapter 14.1. 14.2. 14.3.

14. L-functions L-functions of root systems Proofs of theorems Special values of L-functions

279 280 285 289

Chapter 15.1. 15.2. 15.3. 15.4. 15.5. 15.6. 15.7.

15. Zeta-functions of Lie groups The background and the motivation A general form of zeta-functions Zeta-functions of weight lattices of Lie groups Explicit forms of zeta-functions An application to partial multiple zeta values Functional relations and various evaluation formulas Parity results

293 293 295 298 301 309 311 318

Chapter 16.1. 16.2. 16.3. 16.4. 16.5. 16.6. 16.7. 16.8. 16.9. 16.10.

16. Lattice sums of hyperplane arrangements Notations and the statement of results on lattice sums Examples Proof of Theorem 16.2 The structure of the proof of Theorems 16.4 and 16.5 The generating function and convex polytopes Properties of the polytopes P(m; y) Completion of the proof of Theorems 16.4 and 16.5 A hierarchy and differential equations Proof of Theorems 9.10, 9.11 and 9.12 Proof of Theorem 9.14

323 324 327 333 353 357 359 370 375 377 382

Chapter 17.1. 17.2. 17.3. 17.4. 17.5.

17. Miscellaneous results Functional equations The desingularization The number of representations of SU (3) A connection with Schur multiple zeta-functions Recent results on functional relations

389 389 391 394 395 397

Bibliography

401

Index

413

CHAPTER 1

Introduction The notion of zeta-functions of root systems can be regarded as a certain subclass of the family of multiple zeta-functions. The theory of multiple zeta-functions has a long and rich history, going back to the eighteenth century. We begin with a discussion on various kinds of multiple zeta-functions. In this chapter we mainly focus on the historical account, especially on the history of the theory of meromorphic continuation of multiple zeta-functions. In what follows, N, N0 , Z, Q, R, and C denote the set of all positive integers, non-negative integers, rational integers, rational numbers, real numbers, and complex numbers, respectively. The empty sum is to be understood as 0, and the empty product as 1. 1.1. The Euler double zeta-function The history of the theory of multiple zeta-functions goes back to private communications between C. Goldbach and L. Euler in 1740s. Double zeta values1 (1.1.1)

ζEZ,2 (k1 , k2 ) =

X 1≤n1 1, we have (1.1.4)

ζ(s1 )ζ(s2 ) =

∞ X ∞ X

−s2 1 m−s = 1 m2

m1 =1 m2 =1

X m1 m2

= ζEZ,2 (s1 , s2 ) + ζ(s1 + s2 ) + ζEZ,2 (s2 , s1 ). This was also first discovered by Euler, and is called the harmonic product formula. At first this formula is proved in the region ℜs1 , ℜs2 > 1, but Atkinson’s idea is to use this formula in the case ℜs1 = ℜs2 = 1/2. Therefore Atkinson was forced to consider the meromorphic continuation of ζEZ,2 (s1 , s2 ) appearing on the right-hand side. By using the Poisson summation formula, Atkinson proved a certain integral expression of ζEZ,2 (s1 , s2 ), which enabled him to obtain the meromorphic continuation. Later, Y. Matsuoka [152] proved the continuation of ζEZ,2 (1, s2 ), and Apostol and Vu [5] discussed the continuation of ζEZ,2 (s1 , s2 ) with respect to s1 for each fixed s2 and also with respect to s2 for each fixed s1 . Apostol and Vu also considered the series (1.1.5) ∞ X X

∞ X ∞ X 1 1 = s1 s1 ns2 (m + n) s2 m m (m + m 1 2 ) (2m1 + m2 ) 1 m=1 m 0

(1.2.4)

for all i, j.

Cassou-Nogu`es was inspired by the work of Shintani, and also of Mellin, and wrote a lot of papers (such as [29]) on the meromorphic continuation of ζr (s; P ), its values at negative integer points, and applications to algebraic number theory. Her investigations were succeeded by Sargos, Lichtin, Essouabri and others. So far, multiple zeta-functions defined in this section are in one variable. However multi-variable versions of them may also be considered. Imai [75] and Hida [57] introduced the multi-variable version of Shintani zeta-functions ∞ ∞ Y n X X 1 (1.2.5) ··· , sj (w (m + α ) + · · · + w (m + α )) j1 1 1 jr r r m =0 m =0 j=1 1

r

where (s1 , . . . , sn ) ∈ Cn , and proved the meromorphic continuation to the whole space Cn under the same condition (1.2.4). The meromorphic continuation of more general multi-variable multiple zeta-functions of the form (1.2.6) ζr (s; P1 , . . . , Pn ) =

∞ X m1 =0

···

∞ X mr =0

P1 (m1 , . . . , mr

) s1

1 , · · · Pn (m1 , . . . , mr )sn

where s = (s1 , . . . , sn ), was first studied by Lichtin. Assume that Pj ∈ R[X1 , . . . , Xr ] for 1 ≤ j ≤ n. He [124] proved the meromorphic continuation of ζr (s; P1 , . . . , Pn ) to Cn if each polynomial Pj (1 ≤ j ≤ n) depends explicitly on all variables and is hypoelliptic, that

1.2. MULTIPLE ZETA-FUNCTIONS OF BARNES AND OF MELLIN

5

is, Pj (x1 , . . . , xr ) > 0 for all x = (x1 , . . . , xr ) ∈ [A, ∞)r (with a certain A > 0) and ∂xa11 · · · ∂xarr Pj (1.2.7) (x1 , . . . , xr ) → 0 (1 ≤ j ≤ n) Pj as x ∈ [A, ∞)r , |x| = x1 + · · · + xr → ∞, for all non-negative integers a1 , . . . , ar (not all 0). Then Essouabri [38] [39] proved3 the meromorphic continuation of ζr (s; P1 , . . . , Pn ) under a weaker condition,4 that is, P1 (x) · · · Pn (x) tends to ∞ as |x| → ∞ and (1.2.8) ∂xa11 · · · ∂xarr Pj (x1 , . . . , xr ) is bounded in [A, ∞)r Pj

(1 ≤ j ≤ n).

The method of Essouabri is rather sophisticated, using Hironaka’s desingularization theorem on algebraic varieties [58]. In a different context, the second-named author independently noticed the importance of considering the multi-variable situation. The case r = 2 of the Barnes zeta-function is ∞ X ∞ X 1 (1.2.9) . ζB,2 (s; α, w1 , w2 ) = (m1 w1 + m2 w2 + α)s m =0 m =0 1

2

According to the resemblance of this function and (1.1.9), the double zeta-function in two variables ∞ X ∞ X 1 (1.2.10) s 1 (m1 w1 + α) (m1 w1 + m2 w2 + α)s2 m =0 m =0 1

2

was introduced. This is a generalization of both (1.1.9) and (1.2.9). New asymptotic expansions of the Barnes double zeta-function and related functions were obtained in [130] by analyzing (1.2.10) by the method developed in [91] [92] [93] (mentioned in the preceding section). An important point here is that, considering the two-variable function (1.2.10), we get more flexibility in the analytic treatment, which enables us to obtain a new result. This is the underlying motivation when the second-named author later introduced various multi-variable multiple zeta-functions, such as the Mordell–Tornheim zeta-function (see (1.4.3)) and the four-variable zeta-function associated with the root system B2 (see (1.6.2)). 3In

[39] only the case n = 1 is discussed, but in his thesis [38] it is mentioned that his method can be applied to the multi-variable case. 4This condition is called H S in [38] and [39]. 0

6

1. INTRODUCTION

1.3. Euler–Zagier multiple zeta-functions Another important class of multi-variable multiple zeta-functions is the following generalization of the Euler double zeta-function: (1.3.1)

ζEZ,r (s1 , . . . , sr ) =

X 1≤n1 · · · > ei + er > ei > ei − er ,

and ζr (· ; Dr ) → ζr−1 (· ; Ar−1 ),

(6.4.6) by using (6.4.7)

ei + ei+1 > ei + ei+2 > · · · > ei + er > ei − er−1

(see Figure 6.4.2).

(Br , Cr )

❝

❝

❝

✄✂ ✄✂ ✁✁

❝

❝

❝

❝

(Dr )

❝

❝

❝

✄✂ ✄✂ ✁✁

❝

❝

❝

❝ ❝

Figure 6.4.2. In the cases of Br and Cr , we can also cut off only one of the doubled rightmost edges of the diagram, which gives the relation (6.4.8)

ζr (· ; Br ) → ζr (· ; Ar ),

ζr (· ; Cr ) → ζr (· ; Ar ).

Consider the Br case. The sum mi + · · · + mr , corresponding to the coroot ei + er , exists as one of the factors of ζr (s; Ar ). Hence the final step of (6.4.2) is not necessary this time, and we obtain the first relation of (6.4.8) by following (6.4.9)

2ei > ei + ei+1 > ei + ei+2 > · · · > ei + er .

86

6. THE RECURSIVE STRUCTURE

Similarly, by following ei + ei+1 > ei + ei+2 > · · · > ei + er > ei ,

(6.4.10)

which is (6.4.5) without the last step, we can obtain the second relation of (6.4.8) (see Figure 6.4.3).

(Br , Cr )

❝

❝

❝

✄✂ ✄✂ ✁✁

❝

❝

❝

❝

Figure 6.4.3. In general, we can cut off any edge of the diagram and reduce a zeta-function for a root system to that for another root system. To state our assertion, we recall the definition concerning root systems and construct embeddings in a non-standard sense (that is, embeddings which do not preserve the inner products). Let ∆ be a reduced root system which may not be irreducible, Γ its Dynkin diagram and Ψ = {α1 , . . . , αr } its fundamental system. (The j-th vertex from the left on Γ corresponds to αj .) By Q∨ we denote the coroot lattice generated by Ψ∨ . Let Γ′ be a Dynkin diagram obtained by cutting off some edges from Γ. Let ∆′ , Ψ′ = {α1′ , . . . , αn′ } and (Q′ )∨ be the corresponding root system, fundamental system and coroot lattice, respectively. Then we see that the map f : (Ψ′ )∨ → Ψ∨ defined by f : (αj′ )∨ 7→ αj∨ is Z-linearly extended to an isomorphism f : (Q′ )∨ → Q∨ as Z-modules. P For β ∨ = j cj αj∨ ∈ Q∨ , we denote its height by X (6.4.11) ht β ∨ = cj . j

Lemma 6.10 ([104, Lemma 5.1]). f ((∆′ )∨+ ) ⊂ ∆∨+ . Proof. We show the statement by induction on their heights. We denote by β ∨ ∈ Q∨ the image f ((β ′ )∨ ) of (β ′ )∨ ∈ (∆′ )∨ . We first note that ⟨αi′ , (αj′ )∨ ⟩ ≥ ⟨αi , αj∨ ⟩ because the value of ⟨αi , αj∨ ⟩ is only 0, −1, −2 or −3 for i ̸= j and cutting off some edges of the diagram only produces the effect of increasing the value. Hence in general, we have (6.4.12)

⟨αi′ , (β ′ )∨ ⟩ ≥ ⟨αi , β ∨ ⟩

for (β ′ )∨ ∈ (∆′ )∨+ . Because (β ′ )∨ ∈ (∆′ )∨+ with ht(β ′ )∨ = 1 implies (β ′ )∨ = (αi′ )∨ ∈ (Ψ′ )∨ , for some i, it follows that β ∨ = αi∨ ∈ Ψ∨ by definition.

6.4. RECURSIVE STRUCTURES AND DYNKIN DIAGRAMS

87

Let m ≥ 1, and assume β ∨ ∈ ∆∨+ for (β ′ )∨ ∈ (∆′ )∨+ with ht(β ′ )∨ ≤ P ′ ′ ∨ m. Let (β ′ )∨ = ∈ (∆′ )∨+ with ht(β ′ )∨ = m + 1. It is j cj (αj ) known that there exists a decomposition (β ′ )∨ = (α′ )∨ + (αi′ )∨ with (α′ )∨ ∈ (∆′ )∨+ , ht(α′ )∨ = m and (αi′ )∨ ∈ (Ψ′ )∨ (see, e.g., [70, Section 10.2, Lemma A] or [178, Section 2.11, Proposition A]). If ⟨αi′ , (α′ )∨ ⟩ < 0, then ⟨αi , α∨ ⟩ < 0 by (6.4.12). By the assumption of induction we have α∨ ∈ ∆∨+ and the simple reflection ri (α∨ ) = α∨ − ⟨αi , α∨ ⟩αi∨ is also ∈ ∆∨+ . Hence β ∨ = α∨ + αi∨ ∈ ∆∨+ since all the roots in the αi∨ -string through α∨ belong to ∆∨ . If ⟨αi′ , (α′ )∨ ⟩ ≥ 0, we consider (γ ′ )∨ = ri′ ((β ′ )∨ ), where ri′ is the simple reflection with respect to αi′ . Then (γ ′ )∨ ∈ ∆∨ because (β ′ )∨ ∈ ∆∨ . We see that (γ ′ )∨ = ri′ ((α′ )∨ + (αi′ )∨ ) = (α′ )∨ − (⟨αi′ , (α′ )∨ ⟩ + 1)(αi′ )∨ , which implies that ht(γ ′ )∨ ≤ m − 1. Moreover, we see that (γ ′ )∨ ∈ ∆∨+ ; in fact, since ht(β ′ )∨ = m + 1 ≥ 2, there exists a j ̸= i for which c′j is positive, and so the coefficient of (αj′ )∨ in (γ ′ )∨ is also positive, which implies (γ ′ )∨ ∈ (∆′ )∨+ . Hence by the assumption of induction we have α∨ ∈ ∆∨+ , γ ∨ ∈ ∆∨+ and ri (γ ∨ ) = α∨ + (⟨αi′ , (α′ )∨ ⟩ − ⟨αi , α∨ ⟩ + 1)αi∨ ∈ ∆∨+ . Hence β ∨ = α∨ + αi∨ ∈ ∆∨+ .

□

Theorem 6.11 ([104, Theorem 5.2]). Let ∆, ∆′ be reduced root systems and Γ, Γ′ be their Dynkin diagrams. Then there exists an isomorphism f : (Q′ )∨ → Q∨ such that f ((∆′ )∨+ ) ⊂ ∆∨+ and f : (αj′ )∨ 7→ αj∨ if and only if Γ′ is obtained from Γ by cutting off some edges. Proof. By Lemma 6.10, we have only to show ⟨αi′ , (αj′ )∨ ⟩ ≥ ⟨αi , αj∨ ⟩ if such f exists. Because ⟨αi′ , (αj′ )∨ ⟩ = ⟨αi , αj∨ ⟩ = 2 if i = j, we assume i ̸= j. Since (∆′ )∨+ ∋ ri′ (αj′ )∨ = (αj′ )∨ − ⟨αi′ , (αj′ )∨ ⟩(αi′ )∨ , we have ∆∨+ ∋ f (ri′ (αj′ )∨ ) = αj∨ − ⟨αi′ , (αj′ )∨ ⟩αi∨ . On the other hand, we have ∆∨ ∋ ri αj∨ = αj∨ − ⟨αi , αj∨ ⟩αi∨ . Since the length of the αi∨ -string through αj∨ is ⟨αi , αj∨ ⟩, it follows that the string consists of {αj∨ + hαi∨ | 0 ≤ h ≤ −⟨αi , αj∨ ⟩}. Therefore we have 0 ≥ ⟨αi′ , (αj′ )∨ ⟩ ≥ ⟨αi , αj∨ ⟩. □ Since f is injective, we identify (∆′ )∨+ and its image by f , and denote the image by the same symbol (∆′ )∨+ .

88

6. THE RECURSIVE STRUCTURE

Let (∆∗ )∨ = ∆∨+ \ (∆′ )∨+ and k = #∆∗ . We fix an order (∆∗ )∨ = {β1∨ , β2∨ , . . . , βk∨ }

(6.4.13) by their heights

ht β1∨ ≤ ht β2∨ ≤ · · · ≤ ht βk∨ .

(6.4.14)

For 0 ≤ j ≤ k, define

(∆∗j )∨ = (∆′ )∨+ ∪ {β1∨ , . . . , βj∨ }

(6.4.15) so that

(∆′ )∨+ = (∆∗0 )∨ ⊂ (∆∗1 )∨ ⊂ · · · ⊂ (∆∗k−1 )∨ ⊂ (∆∗k )∨ = ∆∨+ .

(6.4.16)

Then for 1 ≤ j ≤ k we have a decomposition βj∨ = αl∨ + γ ∨ with some αl∨ ∈ Ψ∨ and γ ∨ ∈ (∆∗j−1 )∨ since the order is determined by their heights. Then, again by the Mellin–Barnes argument, we have (6.4.17) ζr (s; ∆∗j ) =

X

Y

λ

α∨ ∈(∆∗j )∨

⟨α∨ , λ⟩−sα



 Y

⟨α∨ , λ⟩−sα  ⟨αl∨ + γ ∨ , λ⟩−sβj

=

X

=

X

Y

λ

∨ α∨ ∈(∆∗j−1 )∨ \{α∨ l ,γ }

 α∨ ∈(∆∗j−1 )∨

λ

×

1 √

Z

2π −1

(c)

⟨α∨ , λ⟩−sα

Γ(sβj + z)Γ(−z) ∨ −sα +z ∨ −sβ −sγ −z ⟨αl , λ⟩ l ⟨γ , λ⟩ j dz, Γ(sβj )

∆∗j

where is the set of positive roots corresponding to (∆∗j )∨ . This implies the following theorem. Theorem 6.12 ([104, Theorem 5.3]). We have Z Γ(sβj + z)Γ(−z) 1 ∗ (6.4.18) ζr (s; ∆j ) = √ Γ(sβj ) 2π −1 (c)

× ζr (. . . , sαl − z, . . . , sβj + sγ + z, . . . ; ∆∗j−1 )dz,

and, repeating this procedure k times, we obtain the recursive relation (6.4.19)

ζr (· ; ∆+ ) → ζr (· ; ∆′+ ).

This general result includes all examples discussed above. Let Γ = Γ(Xr ), X = A, B, C or D. When X = B or C and Γ′ is obtained

6.4. RECURSIVE STRUCTURES AND DYNKIN DIAGRAMS

89

by removing only one of the doubled rightmost edges of Γ, then Γ′ is irreducible. These cases are described as (6.4.8). Except for those cases, any Γ′ which is obtained by cutting off edge(s) of Γ is not irreducible, hence the corresponding zeta-function ζr (· ; ∆′+ ) is the product of two (or more) zeta-functions. If we cut off the leftmost edge, then ζr (· ; ∆′+ ) is the product of ζr−1 (· ; Xr−1 ) and the Riemann zeta-function. These cases are discussed in detail in Sections 6.3 and 6.4. The cases of cutting off the rightmost edge(s) are presented as (6.4.3), (6.4.4) and (6.4.6). More generally, we can cut off the edge which joins two vertices ∨ corresponding to αℓ−1 and αℓ∨ (2 ≤ ℓ ≤ r for X = A, 2 ≤ ℓ ≤ r − 1 for X = B or C, and 2 ≤ ℓ ≤ r − 2 for X = D). Then (6.4.18) implies that ζr (· ; Xr ) can be written as a multiple integral involving ζℓ−1 (· ; Aℓ−1 ) and ζr−ℓ+1 (· ; Xr−ℓ+1 ) (see Figure 6.4.4).

(Ar )

❝

✄✂ ✄✂ ✁✁

❝

αℓ−1 ❝

α❝ℓ

❝

✄✂ ✄✂ ✁✁

❝

❝

(Br , Cr )

❝

✄✂ ✄✂ ✁✁

❝

❝

❝

❝

✄✂ ✄✂ ✁✁

❝

❝

(Dr )

❝

✄✂ ✄✂ ✁✁

❝

❝

❝

❝

✄✂ ✄✂ ✁✁

❝

❝ ❝

Figure 6.4.4. We can summarize the above argument as follows. Theorem 6.13 ([104, Theorem 5.4]). By cutting off any edge of a Dynkin diagram, we find that the zeta-function of the corresponding root system can be written as a (multiple) integral, whose integrand includes zeta-functions of each connected components of the resulting Dynkin diagram. The following diagrams show the hierarchy of Lie algebras whose zeta-functions are connected with each other by Mellin–Barnes recursive formulas as above. The number attached to each algebra is the number of positive roots, that is, the number of variables of the associated zeta-function. The number written in the middle of each arrow is the number of iteration of integrals. The horizontal arrow implies that, in the corresponding process, the Dynkin diagram is not divided

90

6. THE RECURSIVE STRUCTURE

into two separate parts. Note that the following diagrams include the cases of exceptional Lie algebras. Br

/

r2

Ar r(r+1)/2

(ℓ−1)(4r−3ℓ+2)/2

ℓ(r−ℓ+1)





r(r−1)/2

Aℓ−1 ⊕ Br−ℓ+1

Aℓ−1 ⊕

(r−ℓ+1)2

ℓ(ℓ−1)/2

ℓ(ℓ−1)/2

o

Cr

r(r−1)/2

r2

(ℓ−1)(4r−3ℓ+2)/2



Aℓ−1 ⊕ Cr−ℓ+1

Ar−ℓ+1

(r−ℓ+1)(r−ℓ)/2

(r−ℓ+1)2

ℓ(ℓ−1)/2

Dr r(r−1) (r+1)(r−2)/2 ℓ(4r−3ℓ+3)/2



(

Ar−1 ⊕ A1

Aℓ−1 ⊕ Dr−ℓ+1

1

r(r−1)/2

ℓ(ℓ−1)/2

(r−ℓ+1)(r−ℓ)

E8

120

91

A7 ⊕ A1 28

77

81

z

t

D7 ⊕ A1

1

42

96

100

94

1

E 6 ⊕ A2 36





A2 ⊕ A6 3

56

$



A4 ⊕ A4

21

10

3

D5 ⊕ A3

10

20

6

E7 63

41

A6 ⊕ A1 21

32

26

z

t

D6 ⊕ A1

1

30

$

45

47

40

1

36





A2 ⊕ A5 3

E 6 ⊕ A1 

A4 ⊕ A3

15

10

1

D5 ⊕ A2

6

20

3

E6 36

20

z

23



A5 ⊕ A1 15

15

1

20

F4

14

14

14

A2 ⊕ A4

1

/

3

10

A4 10

24



B3 ⊕ A1 9

$

D5 ⊕ A1

1

G2 6

$

C3 ⊕ A1 9

1

2

/

B2 4

*

E7 ⊕ A1 63

1

CHAPTER 7

The meromorphic continuation According to the recursive structure proved in the preceding chapter, we may study the zeta-function of a certain root system of rank r by using the information of zeta-functions of root systems of lower rank. In this chapter, as an example of such reduction philosophy, we will show the meromorphic continuation of zeta-functions of root systems by an induction argument. In this chapter, for two functions f and g, we use Landau’s symbol f = O(g) and Vinogradov’s symbol f ≪ g. Both Landau’s symbol and Vinogradov’s symbol mean that |f (x)| ≤ Cg(x) with a certain constant C > 0 in some indicated region of x. We call C the implied constant.

7.1. The meromorphic continuation of zeta-functions of root systems Using the Mellin–Barnes formula, it is possible to prove the meromorphic continuation of zeta-functions of root systems to the whole complex space. In order to explain the basic idea, we first consider the simplest example, that is the case of ζ2 (s, A2 ), which was already written in the second-named author’s article [131]. At first assume ℜ(sj ) > 1 (1 ≤ j ≤ 3). Applying Lemma 6.1, we have (7.1.1) ζ2 (s; A2 ) =

∞ X ∞ X m1 =1 m2 =1 ∞ X ∞ X

−s2 −s3 1 m−s 1 m2 (m1 + m2 )

m2 −s3 ) m1 m1 =1 m2 =1  z Z ∞ X ∞ X 1 Γ(s3 + z)Γ(−z) m2 −s1 −s3 −s2 √ = m1 m2 dz Γ(s3 ) m1 2π −1 (c) m1 =1 m2 =1 Z ∞ ∞ 1 Γ(s3 + z)Γ(−z) X X −s1 −s3 −z −s2 +z m1 = √ m2 dz Γ(s3 ) 2π −1 (c) m =1 m =1 =

1 −s3 2 m−s m−s 1 2 (1 +

1

2

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Komori et al., The Theory of Zeta-Functions of Root Systems, Springer Monographs in Mathematics, https://doi.org/10.1007/978-981-99-0910-0_7

91

92

7. THE MEROMORPHIC CONTINUATION

=

1 √

Z

2π −1

(c)

Γ(s3 + z)Γ(−z) ζ(s1 + s3 + z)ζ(s2 − z)dz. Γ(s3 )

Here, the two zeta factors in the integrand of the last member of the above are convergent, because −ℜ(s3 ) < c < 0 so ℜ(s1 + s3 + z) = ℜ(s1 ) + ℜ(s3 ) + c > ℜ(s1 ) > 1, ℜ(s2 − z) = ℜ(s2 ) − c > ℜ(s2 ) > 1.

Stirling’s formula (6.1.2) implies that the above integrand is rapidly decreasing when ℑ(z) → ∞. Therefore it is possible to shift the path of integration to the right. We shift the path to ℜ(z) = M − 12 , where M ∈ N. The locations of the poles of the integrand are z = −s3 , −s3 − 1, −s3 − 2, . . . z = 0, 1, 2, . . .

coming from Γ(−z),

z = 1 − s1 − s3 z = s2 − 1

coming from Γ(s3 + z),

coming from ζ(s1 + s3 + z),

coming from ζ(s2 − z).

Therefore the poles relevant to the above shifting of the path are z = s2 − 1 and z = ℓ (0 ≤ ℓ ≤ M − 1). Counting the residues at these poles, we obtain Γ(s2 + s3 − 1)Γ(1 − s2 ) (7.1.2) ζ2 (s; A2 ) = ζ(s1 + s2 + s3 − 1) Γ(s3 )  M −1  X −s3 + ζ(s1 + s3 + ℓ)ζ(s2 − ℓ) ℓ ℓ=0 + IM , where IM =

1 √

2π −1

Z (M − 12 )

Γ(s3 + z)Γ(−z) ζ(s1 + s3 + z)ζ(s2 − z)dz. Γ(s3 )

The poles −s3 − ℓ are not on the path of integration ℜz = M − 1/2 if ℜ(s3 ) > −M + 12 . Similarly, the pole 1 − s1 − s3 is not on the path if ℜ(s1 ) + ℜ(s3 ) > −M + 32 , and the pole s2 − 1 is not on the path if ℜ(s2 ) < M + 12 . Therefore IM is holomorphic in the region  1 3 DM = (s1 , s2 , s3 ) ∈ C ℜ(s3 ) > −M + , 2  3 1 ℜ(s1 ) + ℜ(s3 ) > −M + , ℜ(s2 ) < M + . 2 2 The terms coming from the residues on the right-hand side of (7.1.2) are clearly meromorphic in the whole space C3 . Since M is arbitrary, we may conclude that ζ2 (s; A2 ) can be continued meromorphically to the

7.1. MEROMORPHIC CONTINUATION OF ZETA-FUNCTIONS

93

whole space C3 . Moreover, we can list the candidates of singularities of ζ2 (s; A2 ) from the terms coming from the residues, which are s2 + s3 = 1 − ℓ (ℓ ∈ N0 ) s2 = 1 + ℓ

coming from Γ(1 − s2 ),

s1 + s2 + s3 = 2 s1 + s3 = 1 − ℓ s2 = 1 + ℓ

coming from Γ(s2 + s3 − 1),

coming from ζ(s1 + s2 + s3 − 1),

coming from ζ(s1 + s3 + ℓ),

coming from ζ(s2 − ℓ).

However, we can easily check that two possible singularities coming from two factors (Γ(1 − s2 ) and ζ(s2 − ℓ)) are actually cancelled out. The other candidates in the above list are not cancelled, so they are “true” singularities. Therefore we now arrive at the following: Theorem 7.1 ([131, Theorem 1]). The function ζ2 (s; A2 ) can be meromorphically continued to the whole space C3 , and its singularities are

(7.1.3)

 (ℓ ∈ N0 ),  s1 + s3 = 1 − ℓ s2 + s3 = 1 − ℓ (ℓ ∈ N0 ),  s1 + s2 + s3 = 2.

Remark 7.2. In the case of complex functions of several variables, singularity sets are not isolated points (as in the case of functions of one variable), but hypersurfaces, or more general analytic subsets in the complex space. Typical examples in our present context are (1.3.6), (7.1.3), (7.1.5), and (7.2.1). This produces one of the difficulties in the analytic study of multi-variable multiple zeta-functions. In the above A2 case, it is quite easy to distinguish whether the candidates of singularities, appearing in the Mellin–Barnes integral expression, are indeed singularities, or not. In general, however, it is not so easy. For example, in the case of A3 , we have shown in [146, Theorem 3.5] that the singularities of

(7.1.4)

ζ3 (s; A3 ) =

∞ X ∞ X ∞ X

−s2 −s3 1 m−s 1 m2 m3

m1 =1 m2 =1 m3 =1

× (m1 + m2 )−s4 (m2 + m3 )−s5 (m1 + m2 + m3 )−s6 ,

94

7. THE MEROMORPHIC CONTINUATION

where s = (s1 , s2 , s3 , s4 , s5 , s6 ), are given by  s1 + s4 + s6 = 1 − ℓ (ℓ ∈ N0 ),     s + s + s = 1 − ℓ (ℓ ∈ N0 ), 3 5 6    (ℓ ∈ N0 ),  s2 + s4 + s5 + s6 = 1 − ℓ s1 + s2 + s4 + s5 + s6 = 2 − ℓ (ℓ ∈ N0 ), (7.1.5)   s1 + s3 + s4 + s5 + s6 = 2 − ℓ (ℓ ∈ N0 ),     s + s + s + s + s = 2 − ℓ (ℓ ∈ N0 ),  2 3 4 5 6  s1 + s2 + s3 + s4 + s5 + s6 = 3, and all of these are “true” singularities. In this case, applying Lemma 6.1, we obtain (7.1.6) Z Z 1 Γ(s5 + z)Γ(−z)Γ(s6 + z ′ )Γ(−z ′ ) √ ζ3 (s; A3 ) = Γ(s5 )Γ(s6 ) (2π −1)2 (c) (c′ ) × ζ(s1 , s2 + s5 + z, s4 + s6 + z ′ ; A2 )ζ(s3 − z − z ′ )dz ′ dz, where −ℜs5 < c < 0, −ℜs6 < c′ < 0 ([146, (3.2)]). It is possible to shift the paths of this double integral suitably to show the meromorphic continuation, and then to determine the location of singularities, but the whole argument requires about 20 pages in [146].1 The singularities of zeta-functions of root systems C2 , B3 , C3 are discussed in [104, Section 6], and those of G2 are studied in [110, Section 3]. In the case of C2 , the “true” singularities are completely determined, but in the other cases, only the lists of “possible” singularities are given. A conjecture which predicts the complete list of singularities of zeta-functions of Ar (r ≥ 4) is mentioned at the end of [146, Section 4].2 As we have seen in the proof of Theorem 7.1, to treat the zetafunction of A2 , a vertical shift of the integral is enough. Similarly, vertical shifts work well in the case of C2 , B3 , C3 and G2 . This is because in these cases, the Mellin–Barnes integral expressions correspond to horizontal lines in the diagram given at the end of Section 6.4 (see [104, Remark 6.4]). For example, in [104] we use the integral expression of ζ2 (s; C2 ) involving the zeta-function of A2 , and then in [110] we use the integral expression of ζ2 (s; G2 ) involving the zeta-function of C2 , while the lines between C2 and A2 , and between G2 and C2 , are horizontal in the diagram. 1There

is a misprint in [146, p. 1480]. On l. 11, z5 = 0, 1, . . . , N should be read as z5 = −s5 − l (0 ≤ l ≤ N ). 2In the case of the r-fold Mordell–Tornheim zeta-functions, the true singularities are completely determined for any r ≥ 2 in [140].

7.1. MEROMORPHIC CONTINUATION OF ZETA-FUNCTIONS

95

However, if we have to follow the lines in the diagram which are not horizontal, vertical shifts are not enough. This situation already happened in the study of ζ3 (s; A3 ) in [146]. In this case we have to use more delicate deformation of the path of integration. Actually, such deformation was already introduced in [134], to prove the meromorphic continuation of more general multiple zeta-functions defined by linear forms. Let A = (aij )1≤i≤n,1≤j≤r be an (n, r)-matrix, where each aij ∈ C satisfies ℜaij > 0 or is equal to 0. Put π θi = max {| arg aij |} < . 1≤j≤r 2 aij ̸=0

We assume the following: Assumption 7.3. Each row and each column have at least one nonzero entry. Let y = (y1 , . . . , yr ) ∈ Rr , and define the multiple series associated with y and A by (7.1.7) ζr (s1 , . . . , sn ; y; A ) ∞ ∞ X X √ = ··· e2π −1(m1 y1 +···+mr yr ) (a11 m1 + · · · + a1r mr )−s1 m1 =1

mr =1

× · · · × (an1 m1 + · · · + anr mr )−sn . Under the above assumption, it is clear that this multiple series is absolutely convergent when ℜsj > 1 for all j = 1, . . . , n. Theorem 7.4. The function ζr (s1 , . . . , sn ; y; A ) can be continued meromorphically to the whole space Cn . When y1 = · · · = yr = 0, this result was included in Essouabri’s general theory [38] [39]3 (see Section 1.2), and an alternative proof, based on the Mellin–Barnes formula, was given by the second-named author [134]. The general form of the above theorem was first proved by the first-named author [98]. The proof which we will present in the next section is a generalization of the argument in [134]. An immediate corollary of this theorem is: 3The

[30].

twisted case can also be treated by Essouabri’s method; see de Crisenoy

96

7. THE MEROMORPHIC CONTINUATION

Corollary 7.5. Zeta-functions of root systems (3.1.5) and (3.1.7) can be continued meromorphically to the whole space C#(∆+ ) . In the same paper [134], the meromorphic continuation of multiple zeta-functions of Mordell–Tornheim type and of Apostol–Vu type (see Section 1.4), and also the continuation of ζ2 (s; B2 ) are proved. These cases are obviously special cases of Theorem 7.4, but the proofs in [134] are simpler. The proof in the case of Mordell–Tornheim multiple zeta-functions is based on the recursive structure ζM T,r → ζM T,r−1 → · · · → ζM T,2 = ζ2 (·; A2 ). In the Apostol–Vu case, certain auxiliary multiple series were introduced in [134] to construct a recursive structure, but later Okamoto [167] discovered a modified argument which does not use such auxiliary series. More general multiple series of Mordell–Tornheim type and of Apostol–Vu type are studied in Okamoto [167], Miyagawa [156] and Umezawa [205]. As a preparation to the next section, we conclude this section with the following simple estimations of integrals. Let p, A, B, α, β be real numbers with A + B < 0, and Z ∞ I= (|y| + 1)p exp(A|y + α| + B|y + β|)dy. −∞

Lemma 7.6 ([133, Lemma 3]). We have
(7.1.8) I = O (|α| + 1)p+δ eB|α−β| + (|β| + 1)p+δ eA|α−β| , where δ = 1 or 0 according as A = B or A ̸= B, and the implied constant depends only on p, A and B. Proof. Without loss of generality we may assume α ≥ β. We split the integral I at y = −α and −β to obtain Z −α −Aα−Bβ (7.1.9) I=e (|y| + 1)p e−(A+B)y dy + eAα−Bβ +e =e

Aα+Bβ

−Aα−Bβ

−∞ −β

Z

Z−α ∞

(|y| + 1)p e(A−B)y dy

(|y| + 1)p e(A+B)y dy

−β

I1 + eAα−Bβ I2 + eAα+Bβ I3 ,

say. Integrating by parts (repeatedly if necessary) we get I1 ≪ (|α| + 1)p e(A+B)α ,

I3 ≪ (|β| + 1)p e−(A+B)β ,

7.2. MULTIPLE ZETA-FUNCTIONS DEFINED BY LINEAR FORMS

97

and  I2 ≪ max (|α| + 1)p+δ e−(A−B)α , (|β| + 1)p+δ e−(A−B)β . Substituting these estimates into the right-hand side of (7.1.9), we obtain the desired result. □ Next, define Z ∞ J= (|t + y| + 1)p (|y| + 1)q exp(A|t + y| + B|y|)dy, −∞

where t, q are real and p, A, B are as above. Then: Lemma 7.7 (([133, Lemma 4]). We have
J = O (1 + τ p )τ q+δ eB|t| + (1 + τ p )eA|t| , where τ = |t|+1, δ is as in Lemma 7.6 and the implied constant depends only on p, q, A and B. Proof. Divide Z

Z

J=

= J ∗ + J ∗∗ ,

+ |y|≥2τ

|y| 3 and at first we assume that (s1 , . . . , sn ) is in the region B ∗ = {(s1 , . . . , sn ) | ℜsi > r∗ (1 ≤ i ≤ n)}. Then the series (7.1.7) is absolutely convergent. Rewrite the right-hand side of (7.1.7) as ∞ X m1 =1

···

∞ X

e2π

√

−1(mi y1 +···+mr yr )

mr =1

(a11 m1 + · · · + a1r mr )−s1 · · ·

× (an−1,1 m1 + · · · + an−1,r mr )−sn−1 × (an1 m1 + · · · + an,r−1 mr−1 )−sn  −sn anr mr × 1+ , an1 m1 + · · · + an,r−1 mr−1 and apply Lemma 6.1 with s = sn and λ = anr mr /(an1 m1 + · · · + an,r−1 mr−1 ). Then we have (7.2.4) ζr (s1 , . . . , sn ; y; A ) Z 1 Γ(sn + z)Γ(−z) = √ Γ(sn ) 2π −1 (c) ∞ ∞ X X √ × ··· e2π −1(mi y1 +···+mr yr ) (a11 m1 + · · · + a1r mr )−s1 m1 =1

mr =1

× · · · × (an−1,1 m1 + · · · + an−1,r mr )−sn−1

× (an1 m1 + · · · + an,r−1 mr−1 )−sn −z (anr mr )z dz Z 1 Γ(sn + z)Γ(−z) = √ ζr (s1 , . . . , sn−1 , sn + z, −z; y; A ′ )dz, Γ(s ) 2π −1 (c) n

100

7. THE MEROMORPHIC CONTINUATION

where    A =   ′

a11 · · · a1,r−1 a1r ........................... an−1,1 · · · an−1,r−1 an−1,r an1 · · · an,r−1 0 0 ··· 0 anr

   .  

Here, the choice of c is as follows. To assure the absolute convergence of the multiple series in the integrand, we need ℜsj > 1 (1 ≤ j ≤ n − 1), ℜ(sn + z) > 1, ℜz < −1. These inequalities are satisfied if (s1 , . . . , sn ) ∈ B ∗ and −ℜsn + 1 < c < −1. We choose c under the stronger restriction (7.2.5)

−ℜsn + 2 < c < −1.

This is possible because r∗ > 3. Since ρ(A ′ ) < ρ(A ), we can use the induction assumption to find that ζr (s1 , . . . , sn−1 , sn + z, −z; y; A ′ ) is continued meromorphically, and all the claims of Theorem 7.8 are valid. In particular, possible singularities are of the form (7.2.6)

c1 s1 + · · · + cn (sn + z) − cn+1 z = u(c1 , . . . , cn+1 ) − ℓ,

where c1 , . . . , cn+1 , ℓ ∈ N0 and u(c1 , . . . , cn+1 ) ∈ Z. If cn = cn+1 , then this is (7.2.7)

c1 s1 + · · · + cn sn = u(c1 , . . . , cn ) − ℓ (ℓ ∈ N0 ),

which is irrelevant to z. If cn − cn+1 = d0 > 0, then (7.2.8) −1 z = d−1 0 {−c1 s1 − · · · − cn sn + u(c1 , . . . , cn , cn − d0 )} − d0 ℓ (ℓ ∈ N0 ), and if cn − cn+1 = −e0 < 0, then (7.2.9) −1 z = e−1 0 {c1 s1 + · · · + cn sn − u(c1 , . . . , cn , cn + e0 )} + e0 ℓ (ℓ ∈ N0 ). We write the first term on the right-hand side of (7.2.8) (resp. (7.2.9)) as D(s1 , . . . , sn ; c) (resp. E(s1 , . . . , sn ; c)) for brevity, where c = (c1 , . . . , cn+1 ). Denote the set of all primitive tuples c = (c1 , . . . , cn+1 ) appearing in (7.2.7) (resp. (7.2.8), (7.2.9)) by T0 (resp. TD , TE ). These sets are finite because of Theorem 7.8 (ii). The above (7.2.8) and (7.2.9) can be poles, with respect to z, of the integrand on the right-hand side of (7.2.4). The other poles of the integrand are (7.2.10)

z = −sn − ℓ (ℓ ∈ N0 )

7.2. MULTIPLE ZETA-FUNCTIONS DEFINED BY LINEAR FORMS

101

and z = ℓ (ℓ ∈ N0 ),

(7.2.11)

coming from the gamma factors. We claim that, if r∗ is sufficiently large, then all the poles (7.2.8) and (7.2.10) are on the left of the line ℜz = c, while all the poles (7.2.9) and (7.2.11) are on the right of ℜz = c. This is immediate when (c1 , . . . , cn ) ̸= (0, . . . , 0), because if r∗ is large then ℜsi (1 ≤ i ≤ n) are also large. When c1 = · · · = cn = 0, then cn+1 ̸= 0, and from (7.2.6) we see that the singularities are −1 z = −c−1 n+1 u(c1 , . . . , cn+1 ) + cn+1 ℓ.

(7.2.12)

These singularities form a sequence with the common difference c−1 n+1 ≤ 1. Therefore, if one of them is located on the left of ℜz = c, there should be one of them in the strip c − 1 ≤ ℜz < c. But −ℜsn + 1 < c − 1 by (7.2.5), so this strip is in the domain of absolute convergence of ζr (s1 , . . . , sn−1 , sn + z, −z; y; A ′ ), hence there should not be a singularity in this strip. Therefore all singularities of the form (7.2.12) are on the right of ℜz = c, and hence the claim follows. Now, let (s01 , . . . , s0n ) be any point in the space Cn , and we show that the right-hand side of (7.2.4) can be continued meromorphically to (s01 , . . . , s0n ). First, remove the singularities of the form (7.2.7) from the integrand. These singularities are cancelled by the factor (c1 s1 + · · · + cn sn − u(c1 , . . . , cn ) + ℓ)v(c1 ,...,cn ) (by Theorem 7.8 (iii)). Let L be a sufficiently large positive integer such that, if ℜsi ≥ ℜs0i (1 ≤ i ≤ n), c1 s1 + · · · + cn sn = u(c1 , . . . , cn ) − L does not hold for any c = (c1 , . . . , cn ) ∈ T0 . Define Φ(s1 , . . . , sn ) =

Y L−1 Y

c∈T0 ℓ=0

(c1 s1 + · · · + cn sn − u(c1 , . . . , cn ) + ℓ)v(c1 ,...,cn ) ,

and rewrite (7.2.4) as (7.2.13)

ζr (s1 , . . . , sn ; y; A ) = Φ(s1 , . . . , sn )−1 J(s1 , . . . , sn ),

where (7.2.14)

Γ(sn + z)Γ(−z) Φ(s1 , . . . , sn ) Γ(sn ) 2π −1 c × ζr (s1 , . . . , sn−1 , sn + z, −z; y; A ′ )dz.

J(s1 , . . . , sn ) =

1 √

Z

102

7. THE MEROMORPHIC CONTINUATION

Then the integrand on the right-hand side of (7.2.14) does not have singularities of the form (7.2.7) in the region ℜsi ≥ ℜs0i (1 ≤ i ≤ n). Since Φ(s1 , . . . , sn )−1 is meromorphic in the whole space, in order to complete the proof of the continuation, our remaining task is to show the continuation of J(s1 , . . . , sn ). Let M be a positive integer, and s∗i = s0i + M (1 ≤ i ≤ n). We may choose M so large that (s∗1 , . . . , s∗n ) ∈ B ∗ . Let I1 be the set of all imaginary parts of the poles (7.2.8) and (7.2.10), and I2 be the set of all imaginary parts of the poles (7.2.9) and (7.2.11), for (s1 , . . . , sn ) = (s∗1 , . . . , s∗n ). We divide the discussion into three cases. Case 1. In the case I1 ∩ I2 = ∅, we join D(s∗1 , . . . , s∗n ; c) and D(s01 , . . . , s0n ; c) by the segment S(D; c) which is parallel to the real axis. Similarly, join E(s∗1 , . . . , s∗n ; c) and E(s01 , . . . , s0n ; c) by the segment S(E; c), and join −s∗n and −s0n by the segment S(n). Since I1 ∩ I2 = ∅, we can deform the path ℜz = c to obtain a new path C from c − i∞ to c + i∞, such that all the segments S(D; c) and S(n) are on the left of C, while all the segments S(E; c) and the poles (7.2.11) are on the right of C (see Figure 7.2.1). Then we have Z 1 Γ(sn + z)Γ(−z) (7.2.15) J(s1 , . . . , sn ) = √ Φ(s1 , . . . , sn ) Γ(sn ) 2π −1 C ×ζr (s1 , . . . , sn−1 , sn + z, −z; y; A ′ )dz in a sufficiently small neighborhood of (s∗1 , . . . , s∗n ). Next, on the righthand side of (7.2.15), we move (s1 , . . . , sn ) from (s∗1 , . . . , s∗n ) to (s01 , . . . , s0n ) with keeping the values of imaginary parts of each si . This procedure is possible, because the integrand rapidly decays as |ℑz| tends to +∞ (which is easily seen from the estimate (7.2.2) of ζr (· · · ; y; A ′ ) in the integrand and Stirling’s formula (6.1.2)). During this procedure, the path C does not cross any poles of the integrand. Hence the expression (7.2.15) gives the holomorphic continuation of J(s1 , . . . , sn ) to a neighborhood of (s01 , . . . , s0N ). Case 2. Next, consider the case I1 ∩ I2 ̸= ∅. Then the imaginary part of some member of {D(s∗1 , . . . , s∗n ; c), −s∗n | c ∈ TD } coincides with the imaginary part of some member of {E(s∗1 , . . . , s∗n ; c), 0 | c ∈ TE }. We consider the case (7.2.16)

ℑD(s∗1 , . . . , s∗n ; c1 ) = ℑE(s∗1 , . . . , s∗n ; c2 )

for some c1 and c2 , because other cases can be treated similarly. The ∗ ∗ associated poles are D(s∗1 , . . . , s∗n ; c1 ) − d−1 0 ℓ1 and E(s1 , . . . , sn ; c1 ) + −1 e0 ℓ2 (ℓ1 , ℓ2 ∈ N0 ). When (s∗1 , . . . , s∗n ) is moved to (s01 , . . . , s0n ), these

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103

0 0 poles are moved to D(s01 , . . . , s0n ; c1 ) − d−1 0 ℓ1 and E(s1 , . . . , sn ; c1 ) + −1 e0 ℓ2 , respectively. In the case

(7.2.17)

0 0 −1 ℜD(s01 , . . . , s0n ; c1 ) − d−1 0 ℓ1 ̸= ℜE(s1 , . . . , sn ; c1 ) + e0 ℓ2

for any ℓ1 and ℓ2 , we modify the argument in Case 1 as follows. Let η be a small positive number, and consider the oriented polygonal path S ′ (D; c1 ) joining the points D(s∗1 , . . . , s∗n ; c1 ), D(s∗1 +iη, . . . , s∗n +iη; c1 ), D(s01 + iη, . . . , s0n + iη; c1 ), and then D(s01 , . . . , s0n ; c1 ) in that order. Similarly, define the path S ′ (E; c2 ) which joins E(s∗1 , . . . , s∗n ; c2 ), E(s∗1 + iη, . . . , s∗n +iη; c2 ), E(s01 +iη, . . . , s0n +iη; c2 ), and then E(s01 , . . . , s0n ; c2 ). Then S ′ (D; c1 ) lies on the lower side of the line L = {z | ℑz = ℑD(s∗1 , . . . , s∗n ; c1 ) = ℑE(s∗1 , . . . , s∗n ; c2 )}, while S ′ (E; c2 ) lies on the upper side of L. Because of (7.2.17), we can define the path C ′ , which is almost the same as C, but near the line L we draw C ′ such that it separates [ [ (S ′ (D; c1 ) − d−1 and (S ′ (E; c2 ) + e−1 0 ℓ1 ) 0 ℓ2 ) ℓ1 ∈N0

ℓ2 ∈N0

(see Figure 7.2.2). Then the expression (7.2.15) with C replaced by C ′ is valid in a sufficiently small neighborhood of (s∗1 , . . . , s∗n ). When (s1 , . . . , sn ) moves along the polygonal path joining (s∗1 , . . . , s∗n ), (s∗1 + iη, . . . , s∗n + iη), (s01 + iη, . . . , s0n + iη), and then (s01 , . . . , s0n ) in that order, the path C ′ encounters no pole, hence we obtain the holomorphic continuation. Case 3. The remaining case is that (7.2.18)

0 0 −1 D(s01 , . . . , s0n ; c1 ) − d−1 0 ℓ1 = E(s1 , . . . , sn ; c2 ) + e0 ℓ2

holds for some ℓ1 and ℓ2 . Then this might also hold for some other pairs of (ℓ1 , ℓ2 ). In this case we consider the path C ′′ which is almost the same as C, but near the line L we only require that S(D; c1 ) is on the left of −1 0 0 C ′′ , and that the points E(s∗1 , . . . , s∗n ; c2 )+e−1 0 ℓ2 , E(s1 , . . . , sn ; c2 )+e0 ℓ2 are not on C ′′ for any ℓ2 . When we deform the path ℜz = c on the right-hand side of (7.2.14) to C ′′ , we might encounter several poles of the form (7.2.9). Hence, in a small neighborhood of (s∗1 , . . . , s∗n ), the integral J(s1 , . . . , sn ) has the expression Z 1 Γ(sn + z)Γ(−z) (7.2.19) −R1 (s1 , . . . , sn ) + √ Φ(s1 , . . . , sn ) Γ(sn ) 2π −1 C ′′ ×ζr (s1 , . . . , sn−1 , sn + z, −z; y; A ′ )dz,

104

7. THE MEROMORPHIC CONTINUATION

Figure 7.2.1. The path for Case 1 E(s∗1 , . . . , s∗n )

E(s01 , . . . , s0n )

S(E; c)

−s∗n

−s0n S(n)

c

0 1 2 3

S(D; c) D(s∗1 , . . . , s∗n )

C

D(s01 , . . . , s0n )

where R1 (s1 , . . . , sn ) is the sum of residues of the above poles. Hence R1 (s1 , . . . , sn ) is a (finite) sum of residues which are of the form (7.2.20)

Γ(sn )−1 Φ(s1 , . . . , sn )R(s1 , . . . , sn ; ℓ2 ),

where (7.2.21)  1 dh−1 R(s1 , . . . , sn ; ℓ2 ) = (z − z(ℓ2 ))h Γ(sn + z)Γ(−z) (h − 1)! dz h−1

7.2. MULTIPLE ZETA-FUNCTIONS DEFINED BY LINEAR FORMS

105

Figure 7.2.2. The path for Case 2

C′ E(s01 , . . . , s0n )

D(s01 , . . . , s0n )

 ×ζr (s1 , . . . , sn−1 , sn + z, −z; y; A ) ′

z=z(ℓ2 )

with z(ℓ2 ) = E(s1 , . . . , sn ; c2 ) + e−1 0 ℓ2 , if the order of the pole is h. Note that, as a function in s1 , . . . , sn , R(s1 , . . . , sn ; ℓ2 ) can be continued meromorphically to the whole space Cn . Next, we move (s1 , . . . , sn ) from (s∗1 , . . . , s∗n ) to (s01 , . . . , s0n ); then the path C ′′ might encounter several poles of the same type. Suppose

106

7. THE MEROMORPHIC CONTINUATION

that, during this process, a pole will meet with C ′′ when (s1 , . . . , sn ) = (s•1 , . . . , s•n ), say. In this case, we stop (s1 , . . . , sn ) at (s•1 +ε, . . . , s•n +ε), where ε > 0 is very small, and deform C ′′ slightly to make a new path, of which the pole is on the left. The residue produced by this pole is exactly the same form as (7.2.20), with (s1 , . . . , sn ) = (s•1 +ε, . . . , s•n +ε). But again, this can be continued meromorphically to the whole space. We carry out the same deformation each time when we meet poles, and by R2 (s1 , . . . , sn ) we denote the sum of all residues produced by those poles. When finally we arrive at (s01 , . . . , s0n ), we obtain the following expression: (7.2.22)

J(s1 , . . . , sn ) = −R1 (s1 , . . . , sn ) − R2 (s1 , . . . , sn ) Z 1 Γ(sn + z)Γ(−z) + √ Φ(s1 , . . . , sn ) Γ(sn ) 2π −1 C ′′′ × ζr (s1 , . . . , sn−1 , sn + z, −z; y; A ′ )dz,

where C ′′′ is the path after the above indicated deformation. Since this integral on the right-hand side is holomorphic in a sufficiently small neighborhood of (s01 , . . . , s0n ), this expression gives the meromorphic continuation of J(s1 , . . . , sn ) to (s01 , . . . , s0n ). This completes the proof of the meromorphic continuation4 of ζr (s1 , . . . , sn ; y; A ). Next, we show that all the possible singularity sets of ζr (s1 , . . . , sn ; y; A ) are of the form (7.2.1). This is clear for the singularity sets of Φ(s1 , . . . , sn )−1 . The singularity sets of J(s1 , . . . , sn ) only appear in Case 3. Therefore the point (s01 , . . . , s0n ) on the singularity set satisfies (7.2.18), which can be rewritten as (e0 c11 + d0 c21 )s01 + · · · + (e0 c1n + d0 c2n )s0n

− e0 u(c11 , . . . , c1n , c1n −d0 )+d0 u(c21 , . . . , c2n , c2n +e0 ) = −e0 ℓ1 − d0 ℓ2 ,

where c1 = (c11 , . . . , c1,n+1 ) and c2 = (c21 , . . . , c2,n+1 ). This is of the form (7.2.1). Therefore we obtain Theorem 7.8 (i), (ii) (note that there are only finitely many tuples c, hence also finitely many d0 , e0 ). The assertion of Theorem 7.8 (iii) can be verified by observing the form (7.2.21) and using the induction assumption. Lastly, we show the assertion on the order of ζr (s1 , . . . , sn ; y; A ). First, consider Case 1. Since ζr (s1 , . . . , sn−1 , sn + z, −z; y; A ′ ) satisfies 4If we want to prove only the meromorphic continuation, it is enough to apply the argument in Case 3 to all the cases. However, to prove Theorem 7.8, we have to study more closely when the singularities appear. In fact, Case 1 and Case 2 give the holomorphic continuation. This is the reason why we distinguish the above three cases.

7.2. MULTIPLE ZETA-FUNCTIONS DEFINED BY LINEAR FORMS

107

an estimate of the form (7.2.2) by the induction assumption, the integral on the right-hand side of (7.2.15) clearly satisfies the same type of estimate with respect to ℑs1 , . . . , ℑsn−1 . As for ℑsn , using Stirling’s formula we find that the integral is (7.2.23)Z π  ∞ ≪ exp (|ℑsn | − |ℑsn + y| − |y|) + θn (|ℑsn + y| + |y|) 2 −∞ × F (ℑsn , y)dy,

which is O(eθn |ℑsn | F (ℑsn )) by Lemma 7.7. Hence we obtain the desired assertion in Case 1, and the treatment of Case 2 is similar. In Case 3, we have to estimate R1 (s1 , . . . , sn )+R2 (s1 , . . . , sn ). Since R(s1 , . . . , sn ; ℓ2 ) Z 1 √ = Γ(sn + z)Γ(−z)ζr (s1 , . . . , sn−1 , sn + z, −z; y; A ′ )dz, 2π −1 K where K is a small circle round the point z(ℓ2 ), it is clear that R1 (s1 , . . . , sn ) + R2 (s1 , . . . , sn ) satisfies an estimate of the form (7.2.2) with respect to ℑs1 , . . . , ℑsn−1 . As for ℑsn , there appears the exponential factor which is the same as the exponential factor in the integrand of (7.2.23), so again applying Lemma 7.7 we obtain the desired estimate. Thus we now complete the proof of Theorem 7.4 and Theorem 7.8. □ Remark 7.9. Theorem 7.4 can be generalized to the case of multiple series ∞ ∞ X X (7.2.24) ··· c1 (m1 ) · · · cr (mr )(a11 m1 + · · · + a1r mr )−s1 m1 =1

mr =1

× · · · × (an1 m1 + · · · + anr mr )−sn with arithmetical coefficients c1 (m1 ), . . . , cr (mr ). Besides Assumption 7.3, we further assume that for each cj (mj ) (1 ≤ j ≤ r), the associated P −s Dirichlet series ∞ m=1 cj (mj )mj is absolutely convergent in a certain half-plane ℜs > dj , can be continued meromorphically to C with the only possible pole at s = dj , and is of polynomial order with respect to |ℑs|. Then, by the same argument as above, we can show that the series (7.2.24) can be continued meromorphically to Cn . This is a generalization of the result in [145]. For our later purpose, here we give a variant of Theorem 7.4, where each denominator has a shifting parameter. Let A be as in Theorem 7.4, and let b = (bk )1≤k≤n is an n-component complex vector with

108

7. THE MEROMORPHIC CONTINUATION

ℜbk > 0 (1 ≤ k ≤ n). Define the zeta-function associated with A and b by (7.2.25) ∞ ∞ X X ζr (s1 , . . . , sn ; A , b) = ··· (a11 m1 + · · · + a1r mr + b1 )−s1 m1 =0

mr =0

× · · · × (an1 m1 + · · · + anr mr + bn )−sn . Theorem 7.10 ([105, Theorem 8.1]). The function ζr (s1 , . . . , sn ; A , b) can be continued meromorphically to the whole space Cn . Remark 7.11. The statement of Theorem 8.1√in [105] is a little more general. Choose arbitrary lines ℓi : z = ηe −1θi (η ∈ R) that pass the origin point (1 ≤ i ≤ n). We further choose one of the open half planes partitioned by ℓi , and we denote this half plane by H(ℓi ) for each i. The assumption in [105, Theorem 8.1] is that aij = 0 or aij ∈ H(ℓi ) satisfying Assumption 7.3, while bk are arbitrary. We first prepare the following lemma. Lemma 7.12 ([105, Lemma 8.2]). Let ∞ X (7.2.26) φk (s1 , . . . , sk ) = (a1 m + b1 )−s1 · · · (ak m + bk )−sk m=0

where ℜai > 0, ℜbi > 0 (1 ≤ i ≤ k). Let θi = max{| arg ai |, | arg bi |} < π/2. Then φk (s1 , . . . , sk ) can be continued meromorphically to the whole space Ck , and its possible singularities are located only on s1 + · · · + sk = 1 − l (l ∈ N0 ). Moreover, the estimate (7.2.27) φk (s1 , . . . , sk ) = O(F (ℑs1 , . . . , ℑsk ) exp(θ1 |ℑs1 | + · · · + θk |ℑsk |)) holds uniformly in any fixed vertical multiple strip. Proof. First of all, we divide (7.2.26) as (7.2.28) ∞ X −sk −s1 φk (s1 , . . . , sk ) = b1 · · · bk + (a1 m + b1 )−s1 · · · (ak m + bk )−sk , m=1

and denote the sum on the right-hand side by ψk (s1 , . . . , sk ). Moreover, we define ∞ X ψk∗ (s1 , . . . , sk ) = (a1 m + b1 )−s1 · · · (ak−1 m + bk−1 )−sk−1 m−sk . m=1

It is enough to prove the assertion of the lemma for ψk (s1 , . . . , sk ) instead of φk (s1 , . . . , sk ). For this purpose, we again use Lemma 6.1.

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109

In fact, the following argument is essentially included in [151], in which the case ai = 1 (1 ≤ i ≤ k) is treated. Write σi = ℜsi , ti = ℑsi (1 ≤ i ≤ k). We begin with the situation when σ1 , . . . , σk > 1, so ψk (s1 , . . . , sk ) is absolutely convergent. Since (7.2.29)

(ak m + bk )

−sk

= (ak m)

−sk



bk 1+ ak m

−sk

and | arg(bk /ak m)| < π, we obtain (ak m + bk )

−sk

= (ak m)

−sk

1 √

Z

2π −1

(c)

Γ(sk + z)Γ(−z) Γ(sk )



bk ak m

z dz,

where −σk < c < 0. Therefore we have (7.2.30) ψk (s1 , . . . , sk ) =

∞ X m=1

(a1 m + b1 )−s1 · · · (ak−1 m + bk−1 )−sk−1 (ak m)−sk ×

1 √

Z

Γ(sk + z)Γ(−z) Γ(sk )



2π −1 (c) Z 1 Γ(sk + z)Γ(−z) −sk −z z = √ ak bk Γ(sk ) 2π −1 (c)

bk ak m

z dz

× ψk∗ (s1 , . . . , sk−1 , sk + z)dz.

Here, to ensure the convergence of the ψk∗ factor on the last member of the above, we have to assume that 1 − (σ1 + · · · + σk ) < c < 0. We prove, by induction on k, that ψk∗ (s1 , . . . , sk ) can be continued meromorphically to the whole space Ck , its possible singularities are located only on s1 + · · · + sk = 1 − l (l ∈ N0 ), and the estimate (7.2.31) ψk∗ (s1 , . . . , sk ) = O(F (t1 , . . . , tk ) exp(θ1 |t1 | + · · · + θk−1 |tk−1 |)) holds. When k = 1, we see that ψ1∗ (s1 ) is nothing but the Riemann zeta-function ζ(s1 ), hence the assertion is clear. Let k ≥ 2. Using (7.2.32) (ak−1 m + bk−1 ) = (ak−1 m)

−sk−1

−sk−1

= (ak−1 m) 1 √

2π −1

Z (c′ )

−sk−1



bk−1 1+ ak−1 m

−sk−1

Γ(sk−1 + z ′ )Γ(−z ′ ) Γ(sk−1 )



bk−1 ak−1 m

z ′

dz ′

110

7. THE MEROMORPHIC CONTINUATION

with −σk−1 < c′ < 0, we obtain (7.2.33)

ψk∗ (s1 , . . . , sk )

1 √

=

Γ(sk−1 + z ′ )Γ(−z ′ ) −sk−1 −z′ z′ ak−1 bk−1 Γ(sk−1 )

Z

2π −1 (c′ ) ∗ × ψk−1 (s1 , . . . , sk−2 , sk−1 + sk + z ′ )dz ′ .

∗ To ensure the convergence of the ψk−1 factor on the right-hand side, we have to assume that 1 − (σ1 + · · · + σk ) < c′ < 0. Poles of the integrand on the right-hand side of (7.2.33) are

(7.2.34) (7.2.35) (7.2.36)

z ′ = −sk−1 − l ′

(l ∈ N0 ),

z =l

(l ∈ N0 ),

z ′ = 1 − (s1 + · · · + sk ) − l

(l ∈ N0 ),

∗ which are coming from Γ(sk−1 +z ′ ), Γ(−z ′ ) and ψk−1 (s1 , . . . , sk−2 , sk−1 + ′ sk + z ) under the assumption of induction, respectively. Choose an arbitrarily large M ∈ N and a sufficiently small ε > 0, and shift the path from ℜz ′ = c′ to ℜz ′ = M − 21 . This shifting is possible because, ′ z′ ′ ′ ′ ∗ since a−z k−1 bk−1 = O(exp(2θk−1 |y |)) (where y = ℑz ) and the ψk−1 factor is in the domain of absolute convergence when c′ ≤ ℜz ′ ≤ M − 12 , the integrand is of rapid decay with respect to y ′ . The only relevant poles are at z ′ = l (0 ≤ l ≤ M − 1), with the residue

Γ(sk−1 + l) −s −l ∗ ·Resz′ =l Γ(−z ′ )·ak−1k−1 blk−1 ψk−1 (s1 , . . . , sk−2 , sk−1 +sk +l) Γ(sk−1 )   −sk−1 −sk−1 −l l ∗ =− ak−1 bk−1 ψk−1 (s1 , . . . , sk−2 , sk−1 + sk + l). l

Therefore, from (7.2.33), we have (7.2.37) ψk∗ (s1 , . . . , sk )

M−1 X

 −sk−1 −sk−1 −l l ∗ = ak−1 bk−1 ψk−1 (s1 , . . . , sk−2 , sk−1 +sk +l) l l=0 Z 1 Γ(sk−1 + z ′ )Γ(−z ′ ) −sk−1 −z′ z′ + √ ak−1 bk−1 Γ(sk−1 ) 2π −1 (M − 12 ) ∗ × ψk−1 (s1 , . . . , sk−2 , sk−1 + sk + z ′ )dz ′ .

The integrand on the right-hand side is regular in the region   1 1 . Dk (M ) = (s1 , . . . , sk ) | σk−1 > −M + , σ1 +· · ·+σk > 1−M + 2 2 Therefore (7.2.37) gives the meromorphic continuation of ψk∗ (s1 , . . . , sk ) to Dk (M ). Since we can choose an arbitrarily large M , this implies that ψk∗ (s1 , . . . , sk ) can be continued meromorphically to the whole

7.2. MULTIPLE ZETA-FUNCTIONS DEFINED BY LINEAR FORMS

111

space Ck . Its possible singularities only come from singularities of ∗ ψk−1 (s1 , . . . , sk−2 , sk−1 + sk + l), namely s1 + · · · + sk−2 + (sk−1 + sk + l) = 1 − l′ (l′ ∈ N0 ) from the assumption in the case of k − 1. Moreover, in the region Dk (M ), the integral on the right-hand side of (7.2.37) can be estimated as (7.2.38) Z ∞ π  ≪ exp (|tk−1 | − |tk−1 + y ′ | − |y ′ |) + θk−1 (|tk−1 + y ′ | + |y ′ |) 2 −∞ × F (t1 , . . . , tk , y ′ )dy ′ , which is O(F (t1 , . . . , tk ) exp(θk−1 |tk−1 |)) by Lemma 7.7. Applying this estimate and the k −1 case of (7.2.31) to the right-hand side of (7.2.37), we find that (7.2.31) holds for k. Therefore we have now proved all assertions for ψk∗ (s1 , . . . , sk ). Lastly, we shift the path of integration on the last member of (7.2.30) from ℜz = c to ℜz = M − 12 . Then we obtain  M −1  X −sk −sk −l l ∗ (7.2.39) ψk (s1 , . . . , sk ) = ak bk ψk (s1 , . . . , sk−1 , sk + l) l l=0 Z 1 Γ(sk + z)Γ(−z) −sk −z z + √ ak bk Γ(sk ) 2π −1 (M −ε) × ψk∗ (s1 , . . . , sk−1 , sk + z)dz.

Applying the argument quite similar to the above to (7.2.39), we obtain the assertion of the lemma. □ Proof of Theorem 7.10. We make use of the same method as in the proof of Theorem 7.4: that is, by induction on ρ(A ), we prove the theorem with all assertions of Theorem 7.8 (with ζr (s1 , . . . , sn ; A ) replaced by ζr (s1 , . . . , sn ; A , b)). In the case ρ(A ) = 1, we have ρi = 1 for each i. Hence there exists only one nonzero entry ai,h(i) for each i, and ζr (s1 , . . . , sn ; A , b) =

∞ X m1 =0

···

∞ X

(a1,h(1) mh(1) + b1 )−s1

mr =0

× · · · × (an,h(n) mh(n) + bn )−sn   ∞ Y X Y  = (aij mj + bi )−si  .  1≤j≤r

mj =0

1≤i≤n h(i)=j

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7. THE MEROMORPHIC CONTINUATION

Therefore in this case the assertion of Theorem 7.10 follows from Lemma 7.12. Next, we prove the case ρ(A ) ≥ 2. The condition ρ(A ) ≥ 2 implies that some row of A has at least two nonzero entries. Hence, by renaming the indices, we can assume that an,r−1 ̸= 0 and anr ̸= 0. Then, putting λ= we have

anr mr + 12 bn , an1 m1 + · · · + an,r−1 mr−1 + 12 bn

(an1 m1 + · · · + an,r−1 mr−1 + anr mr + bn )−sn 1 = (an1 m1 + · · · + an,r−1 mr−1 + bn )−sn (1 + λ)−sn 2 1 = (an1 m1 + · · · + an,r−1 mr−1 + bn )−sn 2 Z 1 Γ(sn + z)Γ(−z) z × √ λ dz, Γ(sn ) 2π −1 (c)

where | arg λ| < π because ℜaij > 0 and ℜbn > 0. Therefore we have

ζr (s1 , . . . , sn ; A , b) Z 1 Γ(sn + z)Γ(−z) = √ ζr (s1 , . . . , sn−1 , sn + z, −z; A ′ , b′ )dz, Γ(sn ) 2π −1 (c)

where A ′ is as before and

 b1  ...    n+1  b′ =  b  n−1  ∈ C .  1b  

2 n 1 b 2 n

We see that ℜ( 12 bn ) > 0 and

ρ(A ′ ) = ρ(1)ρ(2) · · · ρ(n − 1) × (ρ(n) − 1) × 1 < ρ(A ).

Hence we can apply the assumption of induction to ζr (s1 , . . . , sn ; A ′ , b′ ). By just the same method as in the proof of Theorem 7.4, we arrive at the assertion for ζr (s1 , . . . , sn ; A , b). This completes the proof of Theorem 7.10. □

CHAPTER 8

Functional relations (I) Now we start to describe our theory on functional relations for zetafunctions of root systems. The original motivation for this theory is Problem 1.1, raised around 2000. When one wants to attack this problem, an obstacle is the fact that the main stream of the research on MZV is based on the Drinfel’d iterated integral expression of MZV. The definition of the Drinfel’d integral will be mentioned in Chapter 13, but the important point here is that the number of iterations of that integral is given by the weight of MZV. It is difficult to extend the definition of such integrals to noninteger values. The third-named author developed an alternative method of studying MZV, which does not depend on iterated integral expressions. Therefore we may expect that his method can be used to consider Problem 1.1. And in fact, the third-named author [203] discovered a new functional relation (Theorem 8.3) by applying his method, which gives the first affirmative answer to Problem 1.1. This method is sometimes called the u-method, because there appears an important parameter u. The third-named author applied this method to evaluate various (multiple) Dirichlet series at positive integers (see [195]–[200]). An interesting feature of the u-method is that the information of trivial zeros of the Riemann zeta-function plays a vital role. The main aim of the present chapter is to explain the umethod and prove Theorem 8.3 and some other functional relations for zeta-functions of root systems. Nakamura [162], inspired by [203], found an alternative proof of Theorem 8.3 (see Remark 8.4). In his proof, Nakamura divided the double sum on the right-hand side of (1.4.1) into several subsums. The first-named author noticed that Nakamura’s division can be explained in terms of Weyl chambers attached to the root system of type A2 . This observation inspired our further theory on functional relations, using the action of Weyl groups. This theory, which will be presented in

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Komori et al., The Theory of Zeta-Functions of Root Systems, Springer Monographs in Mathematics, https://doi.org/10.1007/978-981-99-0910-0_8

113

114

8. FUNCTIONAL RELATIONS (I)

Chapter 9, gives a more general form of functional relations (Theorem 9.11). In Chapter 4, we described a method of evaluating the values of zeta-functions of root systems at even integer points. It is a more difficult problem to evaluate the values at integer points s = (kα )α∈∆+ , where some kα can be odd. It is a remarkable advantage of our theory of functional relations that it catches some methods of evaluating the values of zeta-functions of root systems at such general integer points. Various examples can be found in Example 8.18, Example 8.30, (8.4.2), and Section 9.8. 8.1. A method to evaluate the Riemann zeta-function First, we go back to the well-known Euler formula ζ(2) = π 2 /6. Euler sometimes preferred to consider ∞ X
(−1)n 1−s (8.1.1) ϕ(s) = ϕ(s, 1/2) = = 2 − 1 ζ(s) ns n=1 instead of ζ(s), where ϕ(s, x) is the Lerch zeta-function defined by (4.1.9). Note that ϕ(2) = −π 2 /12. The third member of (8.1.1) is convergent only when ℜs > 0, but Euler essentially found ζ(0) = ϕ(0) = −1/2 and the trivial zeros ζ(−2m) = 0 and ϕ(−2m) = 0 (m ∈ N) by computing divergent series. By the Maclaurin expansions of sin x and cos x, we formally obtain ∞ ∞ X (−1)n sin(nθ) X (−1)j θ2j+1 θ3 = ϕ(2 − 2j) = ϕ(2)θ + , 3 n (2j + 1)! 12 n=1 j=0 ∞ X (−1)n cos(nθ)

n2

n=1

=

∞ X j=0

ϕ(2 − 2j)

(−1)j θ2j θ2 = ϕ(2) + . (2j)! 4

In particular, putting θ = π, we immediately obtain ϕ(2) = −π 2 /12 and ζ(2) = π 2 /6. We begin this chapter by justifying these computations and giving an alternative proof of Euler’s formula for ζ(s) at even positive integers, by considering X (−u)−n (8.1.2) (u ≥ 1) η(s; u) = ns n≥1 instead of ϕ(s). Note that the above series is convergent for ℜs > 0 if u = 1 and for any s ∈ C if u > 1. Obviously η(s; 1) = ϕ(s). Since Z ∞ Z ∞ −x s−1 s Γ(s) = e x dx = n e−nx xs−1 dx (ℜs > 0, n ∈ N), 0

0

8.1. A METHOD TO EVALUATE THE RIEMANN ZETA-FUNCTION

115

we have X  1 n Γ(s)η(s; u) = − e−nx xs−1 dx u 0 n≥1 n Z ∞X Z ∞ 1 xs−1 = − x xs−1 dx = − dx x+1 ue ue 0 0 n≥1 Z

(8.1.3)

∞

(for ℜs > 0, u ≥ 1), which further implies (8.1.4)

η(s; u) =

−1

Γ(s)(e2π

√

−1s

Z

− 1)

C

xs−1 dx, uex + 1

where C denotes the contour consisting of the half-line from +∞ to a small positive number η, a counter-clockwise circle of radius η round the origin, and another half-line from η to +∞. The integral on the right-hand side of (8.1.4) is convergent for all s ∈ C. Therefore (8.1.4) gives the meromorphic continuation of η(s; u) (for u ≥ 1) to the whole complex plane. Moreover, this is holomorphic because (8.1.2) is convergent for ℜs > 0, while in the region ℜs ≤√0, the only possible poles s = −k (k ∈ N0 ), coming from the factor e2π −1s − 1, are cancelled by the gamma factor. Now let us state Euler’s formula: Proposition 8.1. For m ∈ N, (8.1.5)

ζ(2m) =

(−1)m−1 22m−1 π 2m B2m , (2m)!

where Bn is the nth Bernoulli number defined by (4.0.1). The following proof, due to the third-named author [199], is not the simplest proof of (8.1.5), but it includes the essence of the u-method. We prepare some notation and supplemental results as follows. For a small δ > 0 and u ∈ [1, 1 + δ], we set (8.1.6)

F (t; u) =

∞

X 2et tn = E (u) n et + u n=0 n!

(t ∈ C, |t| < π).

(This Taylor expansion is valid for |t| < π, because et + u ̸= 0 in this region.) The coefficients {En (u)} are rational functions in u. In particular,   2et 1 2 F (t; 1) = t =2 1− t + e +1 e − 1 e2t − 1   ∞ 2 t 2t 2 X (1 − 2n )Bn n =2− − = 2 − t , t et − 1 e2t − 1 t n=0 n!

116

8. FUNCTIONAL RELATIONS (I)

which implies (8.1.7)

En (1) =

2(2n+1 − 1)Bn+1 n+1

(n ∈ N),

E0 (1) = 1.

It is noted that E2m (1) = 0 for m ∈ N, which comes from the fact that B2m+1 = 0 for m ∈ N. √ For γ ∈ R with 0 < γ < π, and Cγ : z = γe −1v for 0 ≤ v ≤ 2π, it follows from (8.1.6) that √ Z (2π −1)En (u) −n−1 F (z; u)z dz = (n ∈ N0 ). n! Cγ Since F (t; u) is continuous for (t, u) ∈ Cγ × [1, 1 + δ], we can determine M (γ) =

max

(z,u)∈Cγ ×[1,1+δ]

|F (z, u)|,

which is independent of u ∈ [1, 1 + δ]. Hence we obtain Z |En (u)| 1 (8.1.8) ≤ |F (z; u)| |z|−n−1 |dz| n! 2π Cγ ≤

M (γ) (n ∈ N0 , u ∈ [1, 1 + δ]). γn

Now we claim (8.1.9)

Em (u) = −2η(−m; u)

(m ∈ N0 , u ∈ [1, 1 + δ]).

To prove this claim, we first assume u > 1 and |t| < log u. Then we have X F (t; u) = −2 (−u)−n ent n≥1

= −2

X n≥1

(−u)−n

∞ ∞ X X (nt)m tm = −2 η(−m; u) , m! m! m=0 m=0

where the change of summations is justified by the absolute convergence (note | − u−1 et | < u−1 elog u = 1). Therefore the claim for u > 1 follows. Since both Em (u) and η(−m; u) are continuous in u when u → 1 + 0 (the latter follows from (8.1.4)), the claim is also valid for u = 1. In particular, since η(−m; 1) = (21+m − 1)ζ(−m), the fact E2m (1) = 0 (m ∈ N) is equivalent to ζ(−2m) = 0, the trivial zeros of the Riemann zeta-function. Let us define ∞ X (−u)−m sin(p) (mθ) (8.1.10) Ip (θ; k; u) = , mk m=1

8.1. A METHOD TO EVALUATE THE RIEMANN ZETA-FUNCTION

117

for p ∈ N0 , θ ∈ R, k ∈ N, and u ∈ [1, 1 + δ], where we let dp sin(p) (x) = p sin x. dx The right-hand side of (8.1.10) is convergent for θ ∈ (−π, π) (see [209, §3.35, Example 1]). It is convergent trivially for θ = ±π, except for the case where p is odd, k = 1 and u = 1. It can be easily confirmed that √  √ ( −1)p−1  √−1θ (p) (8.1.11) sin (θ) = e + (−1)p−1 e− −1θ 2 √ ∞ X √ ( −1θ)n p−1 = ( −1) ϵp+1+n , n! n=0 where ϵm = {1 + (−1)m }/2

(8.1.12)

for m ∈ Z. Then we can prove the following lemma ([195, Lemma 2]). Lemma 8.2. (i) Suppose u ∈ (1, 1+δ]. Then Ip (θ; k; u) (p ∈ N0 , k ∈ N) can be analytically continued to θ ∈ C with |θ| < π, and satisfies √ ∞ X √ ( −1θ)n p−1 Ip (θ; k; u) = ( −1) η(k − n; u)ϵp+1+n . n! n=0 (ii) When θ ∈ (−π, π), (8.1.13)

lim Ip (θ; k; u) = Ip (θ; k; 1).

u→1+0

Proof. (i) Since u > 1, we can choose a small ρ > 0 such that u > 1 + ρ. We first assume θ ∈ R which satisfies |θ| < log(1 + ρ). Then we obtain from (8.1.10) and (8.1.11) that (8.1.14)

√ ∞  √ ( −1)p−1 X (−u)−m  m√−1θ p−1 −m −1θ Ip (θ; k; u) = e + (−1) e 2 mk m=1 √ ∞ ∞ X √ (−u)−m X (m −1θ)n p−1 = ( −1) ϵp+1+n , mk n=0 n! m=1

which is absolutely convergent, because ∞ X (e|θ| /u)m < ∞. k m m=1 Changing the order of summation and using (8.1.9), we have √ X  ∞ √ ( −1θ)n p−1 Ip (θ; k; u) = ( −1) η(k − n; u)ϵp+1+n n! n=0

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8. FUNCTIONAL RELATIONS (I)

√ ( −1θ)n = ( −1) η(k − n; u)ϵp+1+n n! n=0 √  ∞ 1X ( −1θ)n − En−k (u)ϵp+1+n . 2 n=k n! √

p−1

X k−1

It follows from (8.1.8) that the right-hand side is absolutely convergent for |θ| < π, which implies that Ip (θ; k; u) can be analytically continued to θ ∈ C with |θ| < π. (ii) We can see that if k ≥ 2 then (8.1.10) is convergent absolutely and uniformly in u ∈ [1, 1 + δ]. Hence we obtain (8.1.13) for k ≥ 2. On the other hand, we should treat the case of k = 1 more carefully. To explain this, we recall the following classical theorem of Abel ([209, §3.71]): P n Let ∞ series whose radius n=0 an z (an ∈ C, n ≥ 0) be a power P of convergence is equal to 1, and assume that ∞ n=0 an converges to α. P∞ n Then limx→1−0 n=0 an x = α. We can easily see that the right-hand side of (8.1.10) with k = 1 is convergent absolutely not only for u ∈ (1, 1 + δ] but also for u ∈ C with |u−1 | < 1. Also (8.1.10) with k = u = 1 is convergent for θ ∈ (−π, π). Therefore it follows from Abel’s theorem that (8.1.13) holds for θ ∈ (−π, π). Thus we complete the proof. □ Define (8.1.15) √

Jp (θ; k; u) = Ip (θ; k; u) − ( −1)

p−1

√ ( −1θ)n η(k − n; u)ϵp+1+n n! n=0

k X

for θ ∈ C with |θ| < π, k ∈ N and u ∈ (1, 1 + δ]. Lemma 8.2 and (8.1.9) imply √ √ ∞ ( −1)p−1 X ( −1θ)n+k (8.1.16) Jp (θ; k; u) = − En (u)ϵp+1+n+k . 2 (n + k)! n=1 Because of (8.1.8), this expression is convergent for u ∈ [1, 1 + δ]. Now let θ ∈ (−π, π), and compute limu→1+0 Jp (θ; k; u) in two ways. Because of (8.1.4), η(k−n; u) (0 ≤ n ≤ k) is continuous in u ∈ [1, 1+δ]. Therefore from (8.1.13) and (8.1.15) we see that (8.1.17) √

lim Jp (θ; k; u) = Ip (θ; k; 1) − ( −1)

u→1+0

p−1

√ ( −1θ)n ϕ(k − n)ϵp+1+n , n! n=0

k X

8.1. A METHOD TO EVALUATE THE RIEMANN ZETA-FUNCTION

119

because η(s; 1) = ϕ(s). On the other hand, the estimate (8.1.8) allows us to change the order of the summation and the limit process u → 1+0 on (8.1.16) to obtain (8.1.18) √ √ ∞ ( −1)p−1 X ( −1θ)n+k lim Jp (θ; k; u) = − En (1)ϵp+1+n+k . u→1+0 2 (n + k)! n=1 Since E2m (1) = 0 (m ∈ N) and ϵm = 0 for odd m, we obtain (8.1.19)

lim Jp (θ; k; u) = 0 (θ ∈ (−π, π))

u→1+0

for k ∈ N and p ∈ N0 with k ̸≡ p (mod 2). Therefore in this case the right-hand side of (8.1.17) is equal to 0. It should be noted that (8.1.19) is the key fact in the proof of Proposition 8.1. We stress that the fact E2m (1) = 0, or equivalently the existence of trivial zeros of the Riemann zeta-function, is essentially used in the proof of (8.1.19). Now we can complete the proof of Proposition 8.1. Note that the delicate case k = 1 in Lemma 8.2 is not necessary in the following proof, but it is prepared for our later purpose. Proof of Proposition 8.1. Let h ∈ N0 and θ ∈ (−π, π). From the above discussion the right-hand side of (8.1.17), with p = 0 and k replaced by 2h + 1, is equal to 0. We then have (8.1.20) I0 (θ; 2h + 1; 1) −

h X j=0

ϕ(2h − 2j)

(−1)j θ2j+1 = 0 (h ∈ N0 ). (2j + 1)!

Now assume h ∈ N. Then I0 (θ; 2h + 1; 1) is continuous1 for θ ∈ [−π, π] (because of the uniform convergence of (8.1.10)) and it tends to 0 as θ → π, so (8.1.21)

h X j=0

ϕ(2h − 2j)

(−1)j π 2j+1 = 0 (h ∈ N). (2j + 1)!

Here we set (8.1.22) (2m)! (2m)! D2m = ϕ(2m) √ = (21−2m − 1)ζ(2m) √ 2m ( −1π) ( −1π)2m 1For

(m ∈ N0 ).

h = 0, the sum on (8.1.20) tends to ϕ(0)π = −π/2 as θ → π, while obviously I0 (π; 1; 1) = 0. Therefore I0 (θ; 1; 1) is not continuous at θ = π.

120

8. FUNCTIONAL RELATIONS (I)

Then (8.1.21) implies that  h  X 2h + 1 D2h−2j = 0 2j + 1 j=0

(h ∈ N).

Since D0 = ζ(0) = −1/2, we obtain !  ∞ h  t X X 2h + 1 t2h+1 D2h−2j − = 2 h=0 j=0 2j + 1 (2h + 1)! ∞ X

∞

t2m X t2j+1 = D2m = (2m)! (2j + 1)! m=0 j=0

∞ X

t2m D2m (2m)! m=0

!

et − e−t 2

for t ∈ C with |t| < π. In fact, we can change the order of summation in the above equation under |t| < π because it follows from (8.1.22) that the last member is convergent absolutely for |t| < π. We see that ∞ X
t tet t t t2m 2m−1 = = − = 1 − 2 B , 2m et − e−t e2t − 1 et − 1 e2t − 1 m=0 (2m)! so we have D2m = (22m−1 − 1)B2m for m ∈ N0 . In view of (8.1.22), we obtain Euler’s formula (8.1.5). □

8.2. Functional relations for the zeta-function of A2 In this section, applying the method similar to that introduced in the previous section, we prove the following functional relation for ζ2 (s1 , s2 , s3 ; A2 ) =

∞ X ∞ X

−s2 −s3 1 m−s 1 m2 (m1 + m2 )

m1 =1 m2 =1

defined by (3.3.19). Theorem 8.3. For k, l ∈ N and s ∈ C, (8.2.1)

ζ2 (k, l, s; A2 ) + (−1)k ζ2 (k, s, l; A2 ) + (−1)l ζ2 (l, s, k; A2 ) [k/2]  X k + l − 2ρ − 1 =2 ζ(2ρ)ζ(s + k + l − 2ρ) l−1 ρ=0  [l/2]  X k + l − 2ρ − 1

ζ(2ρ)ζ(s + k + l − 2ρ), k − 1 ρ=0
where, and in what follows, n0 = 1 for any n ∈ Z. +2

8.2. FUNCTIONAL RELATIONS FOR THE ZETA-FUNCTION OF A2

121

Remark 8.4. This theorem was first proved in a somewhat different form in [203, Theorem 4.5]. In [162],2 Nakamura gave its simpler form (8.2.1) using the method based on Zagier’s technique (see [6, Proposition 4.14]). It can be verified that these two forms are equivalent by an elementary computation (see [103, Theorem 3.1]). Theorem 8.3 gives one of the main motivations to construct the theory of zetafunctions of root systems. From Theorem 8.3, a functional relation involving the Apostol–Vu double zeta-function can be immediately obtained (but has not been published before): Corollary 8.5. For k, l ∈ N and s ∈ C, (8.2.2)

ζAV,2 (k, l, s) + ζAV,2 (l, k, s) + (−1)k {ζAV,2 (k, s, l) + ζAV,2 (s, k, l)}

+ (−1)l {ζAV,2 (l, s, k) + ζAV,2 (s, l, k)} [k/2]  X k + l − 2ρ − 1 =2 ζ(2ρ)ζ(s + k + l − 2ρ) l−1 ρ=0 +2

 [l/2]  X k + l − 2ρ − 1 ρ=0

k−1

ζ(2ρ)ζ(s + k + l − 2ρ)

 − 2−s + (−1)k 2−l + (−1)l 2−k ζ(k + l + s), where ζAV,2 (s1 , s2 , s3 ) is the Apostol–Vu double zeta-function (see (1.4.4)). Proof. It is clear from the definitions that ζ2 (s, t, u; A2 ) = ζAV,2 (s, t, u) + ζAV,2 (t, s, u) + 2−u ζ(s + t + u) (see [131, (5.6)]). Substituting this formula into (8.2.1), we obtain (8.2.2). □ Remark 8.6. Apostol and Vu showed   (−1)a+1 a 1 ζAV,2 (a, a, 1) = (8.2.3) − ζ(2a + 1) 2 4 [a/2]

+ (−1)

a

X ρ=0

ζ(2ρ)ζ(2a + 1 − 2ρ)

for a ∈ N (see [5, (19)]), which was deduced from the functional relation (8.2.4) 2[162]

ζAV,2 (a, s, 1) + ζAV,2 (s, a, 1) was published earlier, but written later than [203].

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8. FUNCTIONAL RELATIONS (I)

  1 = (−1)a+1 ζEZ,2 (1, s + a) + (−1)a+1 + ζ(s + a + 1) 2 a−1 X + (−1)a+1 (−1)m ζ(m + 1)ζ(s + a − m) m=1

for s ∈ C (see [5, (18)]). It is noted that (8.2.3) can be deduced from (8.2.2) with (k, l, s) = (1, a, a), though (8.2.2) and (8.2.4) look different from each other. In the rest of this section, we give the proof of Theorem 8.3. We first prepare several lemmas. Lemma 8.7 ([140, Lemma 2.1]). Let a ∈ N. (i) For an arbitrary function f : N0 → C, a X

(8.2.5)

j=0

ϕ(a − j)ϵa+j

[j/2] X µ=0

f (j − 2µ)

(−1)µ π 2µ (2µ)!

[a/2]

=

X ℓ=0

ζ(2ℓ)f (a − 2ℓ).

(ii) For an arbitrary function g : N0 → C, (8.2.6)

a X j=0

[(j−1)/2]

ϕ(a − j)ϵa+j

X µ=0

g(j − 2µ)

(−1)µ π 2µ 1 = − g(a). (2µ + 1)! 2

Proof. First, notice (8.2.7)

ζ(2h) =

h X µ=0

ϕ(2h − 2µ)

(−1)µ π 2µ (2µ)!

(h ∈ N0 ).

In fact, when h = 0 this is trivial, because ϕ(0) = ζ(0) = −1/2. For h ≥ 1, we can differentiate (8.1.20) with respect to θ and let θ → π to get the result. On the left-hand side of (8.2.5), we change the running index j to ℓ by j = a + 2µ − 2ℓ. By 0 ≤ j ≤ a and 0 ≤ µ ≤ [j/2], we have 0 ≤ 2µ ≤ 2ℓ ≤ a, and hence 0 ≤ ℓ ≤ [a/2]. Therefore, since a − j = 2ℓ − 2µ and j − 2µ = a − 2ℓ, we see that the left-hand side of (8.2.5) is [a/2] ℓ X X (−1)ν π 2ν f (a − 2ℓ) ϕ(2ℓ − 2ν) , (2ν)! ν=0 ℓ=0 which is equal to the right-hand side of (8.2.5) by (8.2.7). Thus we obtain the proof of (8.2.5). Similarly, changing the running indices j to ℓ by j = a + 2µ − 2ℓ on the left-hand side of (8.2.6), and using (8.1.21)

8.2. FUNCTIONAL RELATIONS FOR THE ZETA-FUNCTION OF A2

123

and ϕ(0) = −1/2, we can see that (8.2.6) holds. This completes the proof. □ Corollary 8.8 ([116, Lemma 10.1]). For an arbitrary function f : N0 → C and d ∈ N, we have √ d k X X ( −1π)ν (8.2.8) ϕ(d − k)ϵd−k f (k − ν) ν! ν=0 k=0 √ [d/2] X −1π =− f (d − 1) + ζ(2ξ)ζ(d − 2ξ). 2 ξ=0 √

Proof. Just add two formulas (8.2.5) and (8.2.6) with g(x) = −1πf (x − 1). □

The next lemma was first published in [117, Lemma 5.1], which is a slight modification of [203, Lemma 4.4].3 The origin of this type of result goes back to [195, Lemma 8]. Lemma 8.9. Let {αd }d∈N0 , {βd }d∈N0 , {γd }d∈N0 be sequences of complex numbers such that [d/2]

[d/2]

X (−1)j π 2j (−1)j π 2j αd = γd−2j , βd = γd−2j (2j)! (2j + 1)! j=0 j=0 X

(d ∈ N0 ).

Then [d/2]

(8.2.9)

(8.2.10)

αd = −2

X ν=0

ζ(2ν)βd−2ν

(d ∈ N0 ),

[d/2] 2 X 2ν+2 βd = 2 (2 − 1)ζ(2ν + 2)αd−2ν π ν=0

(d ∈ N0 ),

Proof. First, we prove the following fact: Suppose {ad }d∈N0 , {bd }d∈N0 , and {cd }d∈N0 are sequences of complex numbers such that [d/2]   [d/2]  X d X d+1 (8.2.11) ad = cd−2j , bd = cd−2j (d ∈ N0 ). 2j 2j + 1 j=0 j=0 Then (8.2.12) 3Strictly

 [d/2]  1 X d+1 ad = bd−2ν 22ν B2ν d + 1 ν=0 2ν

(d ∈ N0 ),

speaking, in [203, Lemma 4.4], only the proof of a formula similar to (8.2.9) was given, while a formula similar to (8.2.10) first appeared in [110, Lemma 4.2].

124

8. FUNCTIONAL RELATIONS (I)

where {Bn } are the Bernoulli numbers. In order to prove this fact, we consider the generating function. By (8.2.11), we have  ∞ ∞ X d+1  X X td+1 d+1 td+1 bd = ϵν+1 cd+1−ν (d + 1)! ν (d + 1)! d=0 d=0 ν=0 ∞ X

∞ tν X td+1−ν et − e−t X tµ = ϵν+1 cd+1−ν = cµ . ν! d≥ν−1 (d + 1 − ν)! 2 µ! ν=0 µ=0 P Write 2t/(et − e−t ) = n≥0 ξn tn /(n!), where ξn = 0 for odd n. Then we have  µ  1 X µ+1 bµ−n ξn = cµ µ + 1 n=0 n

for µ ∈ N0 . Therefore from (8.2.11), we have [d/2]   d−2j  X d X d − 2j + 1 1 ad = bd−2j−n ξn . 2j d − 2j + 1 n=0 n j=0 By letting r = d − 2j and ν = 2j + n = d − r + n and exchanging the order of the double sum, (noting r ≡ d (mod 2)) we have     d d X X d 1 r+1 (8.2.13) ad = ϵd+r bd−ν ξν+r−d r r+1 r+ν−d ν=0 r=d−ν  d  ν   X 1 X d+1 ν = bd−ν ϵν+ρ ξρ . d + 1 ν=0 ν ρ ρ=0 We can easily check that

hence

2t et + e−t 2t · = 2t + t, t −t e −e 2 e −1 X ξρ 1 + (−1)η 1 2ν B ν · · = + δ0,1 , ρ! 2 η! ν! ρ+η=ν

where δ0,1 is Kronecker’s delta. Since ξρ = 0 for odd ρ, we obtain ( ν   X ν 2ν B ν (ν : even), ϵν+ρ ξρ = ρ 0 (ν : odd). ρ=0 Substituting this into (8.2.13), we obtain (8.2.12). Now define ad , bd and cd by √ √ √ ( −1π)d ( −1π)d ( −1π)d α d = ad , β d = bd , γ d = cd d! (d + 1)! d!

8.2. FUNCTIONAL RELATIONS FOR THE ZETA-FUNCTION OF A2

125

for d ∈ N0 . Then (8.2.11) holds by the assumption. Hence, by (8.2.12) √ and Euler’s formula ζ(2m) = −22m−1 ( −1π)2m B2m /(2m)! for m ∈ N0 (see (8.1.5)), we complete the proof of (8.2.9). Next, we will prove (8.2.10). Combining the well-known Taylor expansion of cot x: ∞ X (−1)m−1 B2m x cot x = 1 − (2x)2k (2m)! m=1

(see [209, § 7.2]) and (8.1.5), we have πx cot(πx) = −2

(8.2.14)

∞ X

ζ(2m)x2m ,

m=0

and hence πx tan(πx) = −2πx cot(2πx) + πx cot(πx) ∞ X =2 (22m − 1)ζ(2m)x2m .

(8.2.15)

m=1

By (8.2.9) and (8.2.14), we have ∞ X d=0

d

αd x = −2 = −2

∞ d X X

xd

ζ(µ)ϵµ βd−µ

µ=0

d=0 ∞ X

!

! ζ(µ)ϵµ x

µ

µ=0

X

βd−µ x

d−µ

d≥µ

= πx cot(πx)

∞ X

! β m xm

.

m=0

Hence, by (8.2.15), ∞ X d=0

βd x

d

2 = (πx)2 2 = 2 π

∞ X

! (2

m=1

∞ X ℓ=0

2m

− 1)ζ(2m)x

2m

(2

− 1)ζ(ℓ + 2)ϵℓ x

ℓ

∞ X

αn x

! αn x

n

n=0

∞ d 2 X X ℓ+2 = 2 (2 − 1)ζ(ℓ + 2)ϵℓ αd−ℓ π d=0 ℓ=0

Thus we complete the proof of (8.2.10).

! n

n=0

! ℓ+2

∞ X

! xd . □

The following proof of Theorem 8.3 is essentially under the same line as in [203], but here we introduce an additional parameter x.

126

8. FUNCTIONAL RELATIONS (I)

Let r, x, u ∈ R, x ∈ (0, e−π ) and u ∈ [1, 1 + δ] for a sufficiently small δ > 0. We consider several double series associated with u and x. Let ∞ X (−u)−n xn ent (8.2.16) F (t; r, x; u) = . r n n=1 When |t| < π, since |xet | < 1, (8.2.16) is absolutely convergent. Hence F (t; r, x; u) is holomorphic for |t| < π. Using the Taylor expansion of ent , we see that (8.2.17) ∞ ∞ ∞ X X (−u)−n xn X (nt)m tm F (t; r, x; u) = = η(r − m; u/x) . r n m! m! n=1 m=0 m=0 Remark 8.10. In [203], there was no factor xn , so the interchange of the summation as in (8.2.17) was verified in a different way. The argument in [203] is similar to the proof of Lemma 8.2 (i), and the necessary estimate of η(r − m; u) (an analogue of (8.1.8)) was established by using the functional equation for η(s; u) (see [203, Lemma 2.1]). For k ∈ N, set

√ F2 (t; k, r; x; u) = Jk+1 (t/ −1; k; u)F (t; r, x; u),

which is holomorphic for |t| < π, because Jk+1 (t; k; u) is holomorphic in t for |t| < π by (8.1.16) when u ∈ [1, 1 + δ]. Also (8.1.19) implies (8.2.18)

lim F2 (t; k, r; x; u) = 0 (|t| < π).

u→1+0

Remark 8.11. This is the key fact in the proof of Theorem 8.3. In Section 8.1, the formula (8.1.19) gives non-trivial relations (8.1.20) and (8.1.21), from which Proposition 8.1 follows. Here, we again use the √ fundamental fact (8.1.19), but this time we multiply Jk+1 (t/ −1; k; u) by another series F (t; r, x; u) to get a new quantity F2 (t; k, r; x; u). The fact (8.2.18), which is the double analogue of (8.1.19), gives non-trivial relations (Propositions 8.15, 8.16 and 8.17). This is the vital step in our proof of Theorem 8.3. The following double series are necessary to consider the Maclaurin expansion of F2 (t; k, r; x; u). For u ∈ [1, 1 + δ], x ∈ (0, e−π ) and s1 , s2 , s3 ∈ C with ℜs1 , ℜs3 ≥ 1, let ∞ X (−u)−m−n xn , S(s1 , s2 ; s3 ; x; u) = ms1 ns2 (m + n)s3 m,n=1 R1 (s1 , s2 , s3 ; x; u) =

∞ X

(−u)−m−2n xn , ms1 ns2 (m + n)s3 m,n=1

8.2. FUNCTIONAL RELATIONS FOR THE ZETA-FUNCTION OF A2

R2 (s1 , s2 , s3 ; x; u) =

127

∞ X (−u)−2m−n xm+n . ms1 ns2 (m + n)s3 m,n=1

(It can be easily seen that the above series are convergent absolutely4 even if u = 1.) Now assume u ∈ (1, 1 + δ]. Then the above series are convergent absolutely for any s1 , s2 , s3 ∈ C. Using (8.1.15), we find √ (8.2.19) F2 (t; k, r; x; u) = Jk+1 (t/ −1; k; u)F (t; r, x; u) k X √ tj − ( −1)k η(k − j; u)ϵk+j F (t; r, x; u). j! j=0

Apply the second member of (8.1.14) and the definition (8.2.16) to the first term on the right-hand side of the above, while to the second term apply (8.2.17). We obtain F2 (t; k, r; x; u) √ ∞ ( −1)k X (−u)−m−n xn (e(m+n)t + (−1)k e(−m+n)t ) = 2 mk nr m,n=1 k ∞ X √ tj X tm − ( −1)k η(k − j; u)ϵk+j η(r − m; u/x) j! m=0 m! j=0

= F21 − F22 , say. Put j + m = N in F22 to obtain ∞ X k   X √ N tN k F22 = ( −1) η(k − j; u)η(r + j − N ; u/x)ϵk+j , j N! N =0 j=0
where Nj = 0 when N < j. As for F21 , the Taylor expansion of e(m+n)t and e(−m+n)t yields ( ∞ √ ∞ ( −1)k X (−u)−m−n xn X (m + n)N tN F21 = 2 mk nr N! m,n=1 N =0 ! ) ∞ X X X (−u)−m−n xn (−1)k X (−m + n)N tN + + + . mk nr N! m>n mn , put m − n = l to find that X m>n 4Actually

convergence.

=

∞ ∞ X (−u)−l−2n xn (−1)k X (−l)N tN (n + l)k nr N! N =0 l,n=1

a weaker condition ℜ(s1 + s3 ) > 1 is enough to ensure the absolute

128

8. FUNCTIONAL RELATIONS (I)

=

∞ X

(−1)k+N R1 (−N, r, k; x; u)

N =0

Similarly, the sum

P

m 0, θ ∈ R and b, c ∈ N0 , we have (8.2.22) √    [(b+c)/2]  c X X (−θ)ν b+c−ν cos(ν) (θx) b+c−2N ( −1θ)2N = . ν! c−ν xb+1+c−ν c xb+1+c−2N (2N )! ν=0 N =0 Proof. Let

(−1)c cos(θx) · . b+1 c! Px∞ √ Insert the Taylor expansion cos(θx) = N =0 ( −1θx)2N /(2N )! to the right-hand side and differentiate c times with respect to x to obtain √ ∞ (−1)c X ( −1θ)2N (c) f (x) = (2N − b − 1) · · · (2N − b − c)x2N −b−c−1 c! N =0 (2N )! f (x) =

8.2. FUNCTIONAL RELATIONS FOR THE ZETA-FUNCTION OF A2 [(b+c)/2] 

=

X

N =0

129

√  b + c − 2N ( −1θ)2N . c xb+1+c−2N (2N )!

On the other hand, we have c   (−1)c X c (c) f (x) = (cos(θx))(ν) (x−b−1 )(c−ν) c! ν=0 ν c X

θν cos(ν) (θx)(−b − 1) · · · (−b − c + ν)x−b−c+ν−1 ν!(c − ν)! ν=0   c X (−θ)ν b + c − ν cos(ν) (θx) = , ν! c−ν xb+c+1−ν ν=0 = (−1)c

hence the lemma.

□

Let b, c ∈ N0 and θ, r ∈ R. Put x = m in (8.2.22), multiply by (−u)−m m−r and sum up over all m ∈ N to get   ∞ c X (−θ)ν b + c − ν X (−u)−m cos(ν) (mθ) (8.2.23) ν! c−ν mb+1+c+r−ν ν=0 m=1 √  ∞ [(b+c)/2]  X b + c − 2N X (−u)−m ( −1θ)2N = . c mb+1+c+r−2N (2N )! m=1 N =0 This leads to the following lemma. Lemma 8.14. Let k, d ∈ N, r ∈ R, θ ∈ (−π, π), x ∈ (0, e−π ) and u ∈ (1, 1 + δ]. Then ∞  1 X (−u)−m−n xn cos((m + n)θ) (8.2.24) 2 m,n=1 mk nr (m + n)d (−u)−m−2n xn cos(mθ) md nr (m + n)k  −2m−n m+n x cos(nθ) k (−u) + (−1) mk nd (m + n)r   j k X X (−θ)ν d − 1 + j − ν j − η(k − j; u)(−1) ϵk+j ν! j−ν ν=0 j=0 + (−1)k+d

∞ X (−u)−m xm cos(ν) (mθ) md+j+r−ν m=1 √ ∞ X ( −1θ)2N = Ed−2N,2 (k, r; x; u) . (2N )! N =0

×

130

8. FUNCTIONAL RELATIONS (I)

Proof. Since u ∈ (1, 1 + δ], using the Taylor expansion of cos(·) we can see that √ ∞ ∞ X (−u)−m−n xn cos((m + n)θ) X ( −1θ)2N = S(s1 , s2 , s3 − 2N; x; u) , s1 ns2 (m + n)s3 m (2N)! m,n=1 N =0 √ ∞ ∞ −m−2n n X X (−u) x cos(mθ) ( −1θ)2N = R1 (s1 − 2N, s2 , s3 ; x; u) , ms1 ns2 (m + n)s3 (2N)! m,n=1 N =0 √ ∞ ∞ X (−u)−2m−n xm+n cos(nθ) X ( −1θ)2N = R2 (s1 , s2 − 2N, s3 ; x; u) . ms1 ns2 (m + n)s3 (2N)! m,n=1 N =0 Furthermore, by applying (8.2.23) with (b, c) = (d − 1, j), we have   ∞ j X (−θ)ν d − 1 + j − ν X (−u)−m xm cos(ν) (mθ) (−1) ν! j−ν md+j+r−ν ν=0 m=1 √  ∞  X 2N − d ( −1θ)2N = η(d + j + r − 2N ; u/x) , j (2N )! N =0 j

because 

d − 1 + j − 2N j

 = (−1)

j



 2N − d . j

Therefore, by (8.2.20), we obtain the assertion.

□

Similar to the case of En (u) in the preceding section, from (8.2.21) we can confirm that for any γ ∈ (0, π) there exists M2 (γ) > 0 depending on (k, r, x) but independent of u ∈ [1, 1 + δ] such that Z |E−N,2 (k, r; x; u)| 1 (8.2.25) ≤ |F2 (z; k, r, x; u)| |z|−N −1 |dz| N! 2π Cγ ≤

M2 (γ) (N ∈ N). γN

This means that the right-hand side of (8.2.24) is uniformly convergent in u ∈ [1, 1 + δ]. Hence we can let u → 1 on both sides of (8.2.24), namely (8.2.24) holds for u = 1 and θ ∈ (−π, π). It follows from (8.2.18) and (8.2.21) that (8.2.26) ( P −m −k−r (−1)k+1 2−1 ∞ m (N = 0) m=1 x lim E−N,2 (k, r; x; u) = u→1+0 0 (N ≥ 1). Therefore we have the following result.

8.2. FUNCTIONAL RELATIONS FOR THE ZETA-FUNCTION OF A2

131

Proposition 8.15. Let k, d ∈ N, r, θ, x ∈ R, θ ∈ (−π, π) and x ∈ (0, e−π ). Then ∞  1 X (−1)m+n xn cos((m + n)θ) (8.2.27) 2 m,n=1 mk nr (m + n)d (−1)m xn cos(mθ) md nr (m + n)k  n m+n cos(nθ) k (−1) x + (−1) mk nd (m + n)r   j k X X (−θ)ν d − 1 + j − ν j − ϕ(k − j)(−1) ϵk+j ν! j−ν ν=0 j=0 + (−1)k+d

∞ X (−1)m xm cos(ν) (mθ) × md+j+r−ν m=1 √ [d/2] X ( −1θ)2N = Ed−2N,2 (k, r; x; 1) . (2N )! N =0

When d ≥ 2, we can differentiate (8.2.27) with respect to θ because of its uniform convergency. Using the known relation       x−1 x x−1 − + = , y−1 y y we have the following.

Proposition 8.16. Let k, d ∈ N with d ≥ 2, r, θ, x ∈ R, θ ∈ (−π, π) and x ∈ (0, e−π ). Then ∞  1 X (−1)m+n xn sin((m + n)θ) (8.2.28) − 2 m,n=1 mk nr (m + n)d−1 (−1)m xn sin(mθ) md−1 nr (m + n)k  n m+n sin(nθ) k (−1) x + (−1) mk nd−1 (m + n)r + (−1)k+d

−

k X j=0

×

j

ϕ(k − j)(−1) ϵk+j

X N =1

ν=0

ν!

∞ X (−1)m xm cos(ν+1) (mθ) md−1+j+r−ν m=1

[d/2]

=

  j X (−θ)ν d − 2 + j − ν

Ed−2N,2 (k, r; x; 1)

(−1)N θ2N −1 . (2N − 1)!

j−ν

132

8. FUNCTIONAL RELATIONS (I)

From these results, we consequently give the proof of Theorem 8.3. We give a more general result. Proposition 8.17. For k, l ∈ N, s, x ∈ C with |x| < 1, we have ∞  X xn xn k+l (8.2.29) + (−1) mk ns (m + n)l ml ns (m + n)k m,n=1  xm+n k + (−1) k l m n (m + n)s   ∞ [k/2] X k + l − 1 − 2ρ X xm k = 2(−1) ζ(2ρ) l−1 mk+l+s−2ρ ρ=0 m=1 [l/2] X

 ∞ k + l − 1 − 2ρ X xm + 2(−1) ζ(2ρ) . k−1 mk+l+s−2ρ ρ=0 m=1 k



Proof of Proposition 8.17. Let k, d ∈ N, r, x ∈ R with x ∈ (0, e−π ). Both sides of (8.2.27) and (8.2.28) are continuous in θ for θ ∈ [−π, π]. Letting θ → π in (8.2.27) we obtain ∞  1 X xn xn k+d (8.2.30) + (−1) 2 m,n=1 mk nr (m + n)d md nr (m + n)k  xm+n k + (−1) k d m n (m + n)r −

k X

ϕ(k − j)(−1)j ϵk+j

[j/2] X (−1)µ π 2µ

(2µ)!   ∞ d − 1 + j − 2µ X xm × j − 2µ md+j+r−2µ m=1 √ [d/2] X ( −1π)2N = Ed−2N,2 (k, r; x; 1) , (2N )! N =0 j=0

µ=0

and similarly, replacing d by d + 2 in (8.2.28), dividing by −π and letting θ → π, we obtain (8.2.31)

k X j=0

[(j−1)/2]

X (−1)µ π 2µ (2µ + 1)! µ=0   ∞ d − 1 + j − 2µ X xm × j − 2µ − 1 md+j+r−2µ m=1 j

ϕ(k − j)(−1) ϵk+j

[(d+2)/2]

=

X N =1

Ed+2−2N,2 (k, r; x; 1)

(−1)N −1 π 2N −2 (2N − 1)!

8.2. FUNCTIONAL RELATIONS FOR THE ZETA-FUNCTION OF A2 [d/2]

=

X N =0

Ed−2N,2 (k, r; x; 1)

133

(−1)N π 2N . (2N + 1)!

Denote by αd and βd the left-hand sides of (8.2.30) and (8.2.31), respectively. Then we may apply Lemma 8.9 (with γd = Ed,2 (k, r; x; 1)) to obtain [d/2]

αd = −2

(8.2.32)

X

βd−2ρ ζ(2ρ).

ρ=0

Moreover, by Lemma 8.7, the triple sums on the left-hand sides of (8.2.30) and of (8.2.31) are   ∞ [k/2] X d − 1 + k − 2ρ X xm k (−1) ζ(2ρ) , k − 2ρ md+k+r−2ρ ρ=0 m=1 and   ∞ (−1)k d + k − 1 X xm , d+k+r 2 k−1 m m=1 respectively. From these expressions and (8.2.32), we have ∞  X xn xn k+d (8.2.33) + (−1) mk nr (m + n)d md nr (m + n)k m,n=1  xm+n k + (−1) k d m n (m + n)r   ∞ [k/2] X k + d − 1 − 2ρ X xm k = 2(−1) ζ(2ρ) d−1 md+k+r−2ρ ρ=0 m=1 [d/2]

 ∞ k + d − 1 − 2ρ X xm + 2(−1) ζ(2ρ) . k−1 md+k+r−2ρ ρ=0 m=1 k

X



Replacing d and r by l and s, respectively, we obtain (8.2.29) for x ∈ (0, e−π ) and s ∈ R. Since both sides of (8.2.29) are convergent absolutely for all x, s ∈ C with |x| < 1, we arrive at the assertion. □ The above Lemma 8.14 and Propositions 8.15, 8.16, 8.17 are slight generalizations (that is, with an additional factor xn ) of the results included in [203]. Now we are ready to complete the proof of Theorem 8.3. Proof of Theorem 8.3. Temporarily assume ℜs ≥ 1. Then (8.2.29) holds for |x| = 1 because both sides are convergent uniformly

134

8. FUNCTIONAL RELATIONS (I)

for |x| ≤ 1. Therefore, putting x = e2π sides by (−1)k , we obtain

√

−1y

(y ∈ R) and dividing both

(8.2.34) ζ2 (k, l, s; (y, y); A2 ) + (−1)k ζ2 (k, s, l; (1, y); A2 ) + (−1)l ζ2 (l, s, k; (1, y); A2 ) [k/2]  X k + l − 2ρ − 1 =2 ζ(2ρ)ϕ(s + k + l − 2ρ, y) l−1 ρ=0 +2

 [l/2]  X k + l − 2ρ − 1 ρ=0

k−1

ζ(2ρ)ϕ(s + k + l − 2ρ, y),

where ϕ(s, y) is defined by (4.1.9). Since all functions on both sides are continued meromorphically (as functions in s) to C (see Theorem 7.1 and Corollary 7.5), (8.2.34) is valid for any s ∈ C. In particular, putting y = 0, we obtain the assertion of Theorem 8.3. □ Example 8.18. Setting (k, l) = (2, 2), (3, 2) in (8.2.1), we have (8.2.35) (8.2.36)

ζ2 (2, 2, s; A2 ) + 2ζ2 (2, s, 2; A2 ) = 4ζ(2)ζ(s + 2) − 6ζ(s + 4),

ζ2 (3, s, 2; A2 ) − ζ2 (3, 2, s; A2 ) − ζ2 (2, s, 3; A2 ) = 10ζ(s + 5) − 6ζ(2)ζ(s + 3).

Setting s = 2 in (8.2.35) and (8.2.36), we have (8.2.37) (8.2.38)

1 6 π , 2835 ζ2 (2, 2, 3; A2 ) = 6ζ(2)ζ(5) − 10ζ(7), ζ2 (2, 2, 2; A2 ) =

respectively, where (8.2.37) was given by Mordell [157] and (8.2.38) was given by Tornheim [194]. Note that ζ2 (k, 0, l; A2 ) = ζEZ,2 (k, l). Then, setting s = 0 in (8.2.36), we have ζEZ,2 (3, 2) − ζEZ,2 (2, 3) = 10ζ(5) − 5ζ(2)ζ(3). On the other hand, a special case of (1.1.4) gives ζ(2)ζ(3) = ζEZ,2 (3, 2) + ζEZ,2 (2, 3) + ζ(5). Combining these results, we recover the known results 9 ζEZ,2 (3, 2) = ζ(5) − 2ζ(2)ζ(3), 2 11 ζEZ,2 (2, 3) = − ζ(5) + 3ζ(2)ζ(3) 2 (cf. [15]). Therefore this example gives an (chronologically first) affirmative answer to Problem 1.1.

8.2. FUNCTIONAL RELATIONS FOR THE ZETA-FUNCTION OF A2

135

Remark 8.19. We introduced the additional factor xn in the above proof. There are several advantages; first, it makes the convergence argument simpler (see Remark 8.10). Secondly, it produces a slightly generalized result with twisted factors such as (8.2.34). However, the original motivation of introducing such a factor was different. When we first applied the idea of u-method to zeta-functions of higher ranks (e.g. [146]), it only produced functional relations in which some of the variables should be equal to 0. In order to remove this unsatisfactory point, we introduced the additional factor such as xn on the numerators and consider the (repeated) integration with respect to x to obtain more general functional relations (see [149] [100]). Later, this idea was refined to the form of Lemma 8.24, which will be proved in the next section. In fact, it can be seen that√the numerators in the statement of Lemma 8.24 include the factor eN −1θ , and repeated integration with respect to θ is performed in its proof. (See also Remark 15.2.) It should be noted that Theorem 8.3 contains the following fact which is equivalent to [189, Theorem 4.1] given by Subbarao and Sitaramachandrarao. However, since their result includes several misprints, we correct them as follows. Corollary 8.20. For m, n, p ∈ N, (8.2.39) ζ2 (2m, 2n, 2p; A2 ) + ζ2 (2m, 2p, 2n; A2 ) + ζ2 (2n, 2p, 2m; A2 )  m  X 2m + 2n − 2ρ − 1 =2 ζ(2ρ)ζ(2m + 2n + 2p − 2ρ) 2n − 1 ρ=0  n  X 2m + 2n − 2ρ − 1 +2 ζ(2ρ)ζ(2m + 2n + 2p − 2ρ). 2m − 1 ρ=0 In particular, (8.2.40)  m  4 X 4m − 2ρ − 1 ζ2 (2m, 2m, 2m; A2 ) = ζ(2ρ)ζ(6m − 2ρ). 3 ρ=0 2m − 1 Examples listed at the end of Section 4.2 can now be immediately deduced from (8.2.40). Also, as a corollary of Theorem 8.3, we have the following fact which implies a kind of “parity result” (see Section 12.5). This result was first given by Tornheim [194] (see also [69]). Corollary 8.21. For a, b, c ∈ N with 2 ∤ (a + b + c), (8.2.41)

ζ2 (a, b, c; A2 ) ∈ Q[π 2 , {ζ(2j + 1)}j∈N ].

136

8. FUNCTIONAL RELATIONS (I)

Proof. When 2 ∤ (a + b + c), a, b, c are all odd, or one of them is odd and the other two are even. As for the former case, setting (k, l, s) = (2p + 1, 2q + 1, 2r + 1) and (k, l, s) = (2p + 1, 2r + 1, 2q + 1) in (8.2.1), we see that the left-hand sides are equal to ζ2 (2p + 1, 2q + 1, 2r + 1; A2 ) − ζ2 (2p + 1, 2r + 1, 2q + 1; A2 ) − ζ2 (2q + 1, 2r + 1, 2p + 1; A2 ),

ζ2 (2p + 1, 2r + 1, 2q + 1; A2 ) − ζ2 (2p + 1, 2q + 1, 2r + 1; A2 ) − ζ2 (2r + 1, 2q + 1, 2p + 1; A2 ),

respectively. By adding them and noting ζ2 (a, b, c; A2 ) = ζ2 (b, a, c; A2 ), we have ζ2 (2r + 1, 2q + 1, 2p + 1; A2 ) ∈ Q[π 2 , {ζ(2j + 1)}j∈N ] (p, q, r ∈ N0 ). As for the latter case, for example, setting (k, l, s) = (2p + 1, 2q, 2r) and (k, l, s) = (2p + 1, 2r, 2q) in (8.2.1), we see that the left-hand sides are equal to ζ2 (2p + 1, 2q, 2r; A2 ) − ζ2 (2p + 1, 2r, 2q; A2 ) + ζ2 (2q, 2r, 2p + 1; A2 ), ζ2 (2p + 1, 2r, 2q; A2 ) − ζ2 (2p + 1, 2q, 2r; A2 ) + ζ2 (2r, 2q, 2p + 1; A2 ), respectively. Hence we have ζ2 (2r, 2q, 2p + 1; A2 ) ∈ Q[π 2 , {ζ(2j + 1)}j∈N ] (p ∈ N0 , q, r ∈ N). As for other cases, we can prove them in the same way.

□

Example 8.22. Here we use the method introduced in the proof of Corollary 8.21, that is, combining functional relations in (8.2.1). Then we can obtain ζ2 (1, 1, 1; A2 ) = 2ζ(3), 1 ζ2 (1, 1, 3; A2 ) = − ζ(3)π 2 + 4ζ(5), 3 1 3 ζ2 (1, 2, 2; A2 ) = ζ(3)π 2 − ζ(5), 6 2 1 2 ζ2 (1, 3, 1; A2 ) = − ζ(3)π + 3ζ(5), 6 1 3 ζ2 (2, 1, 2; A2 ) = ζ(3)π 2 − ζ(5), 6 2 1 2 ζ2 (2, 2, 1; A2 ) = ζ(3)π − 3ζ(5), 3 1 ζ2 (3, 1, 1; A2 ) = − ζ(3)π 2 + 3ζ(5), 6 1 1 ζ2 (1, 1, 5; A2 ) = − ζ(3)π 4 − ζ(5)π 2 + 6ζ(7), 45 3

8.3. FUNCTIONAL RELATIONS FOR THE ZETA-FUNCTION OF C2

137

1 ζ2 (1, 2, 4; A2 ) = ζ(5)π 2 − 5ζ(7), 2 7 ζ2 (1, 3, 3; A2 ) = − ζ(5)π 2 + 12ζ(7), 6 1 1 ζ2 (1, 4, 2; A2 ) = ζ(3)π 4 + ζ(5)π 2 − 6ζ(7), 90 2 1 1 ζ2 (1, 5, 1; A2 ) = − ζ(3)π 4 − ζ(5)π 2 + 4ζ(7), 90 6 1 2 ζ2 (2, 1, 4; A2 ) = ζ(5)π − 5ζ(7), 2 ζ2 (2, 2, 3; A2 ) = ζ(5)π 2 − 10ζ(7), 1 ζ2 (2, 3, 2; A2 ) = − ζ(5)π 2 + 2ζ(7), 6 1 1 ζ2 (2, 4, 1; A2 ) = ζ(3)π 4 + ζ(5)π 2 − 4ζ(7), 90 3 7 2 ζ2 (3, 1, 3; A2 ) = − ζ(5)π + 12ζ(7), 6 1 ζ2 (3, 2, 2; A2 ) = − ζ(5)π 2 + 2ζ(7), 6 1 ζ2 (3, 3, 1; A2 ) = − ζ(5)π 2 + 4ζ(7), 3 1 1 ζ2 (4, 1, 2; A2 ) = ζ(3)π 4 + ζ(5)π 2 − 6ζ(7), 90 2 1 1 ζ2 (4, 2, 1; A2 ) = ζ(3)π 4 + ζ(5)π 2 − 4ζ(7), 90 3 1 1 4 ζ2 (5, 1, 1; A2 ) = − ζ(3)π − ζ(5)π 2 + 4ζ(7). 90 6 The symmetric property ζ2 (a, b, c; A2 ) = ζ2 (b, a, c; A2 ) can be easily read off from the above. In fact, this follows from the definition (3.3.19). For more details, see Section 9.3. Remark 8.23. Functional relations which connect ζ3 (s; A3 ) with ζ2 (s; A2 ) were given by the authors [146, Theorems 5.9, 5.10] [100, Theorem 3.4] and also by Nakamura [163, Theorem 6.2]. Then, functional relations which connect ζ3 (s; A3 ) with ζ(s), ϕ(s) were first obtained in [103, Theorem 7.1] for the case of even integral arguments, and later in [114, Theorem 9] for the more general case, by the same principle as in the present section. See Remark 15.26. 8.3. Functional relations for the zeta-function of C2 By using the u-method explained in the previous section, we can obtain functional relations for zeta-functions of root systems of other

138

8. FUNCTIONAL RELATIONS (I)

types, for instance type A3 (see [146] [100] [103] [114]), types C2 , B3 , C3 (see [103]) and type G2 (see [110] [117]). In the previous section, three lemmas of elementary nature (Lemmas 8.7, 8.9 and 8.13) played important roles. In particular, Lemma 8.9 (and its primitive form [195, Lemma 8]) was indispensable in the early stage of the u-method (such as [203] [146] [140] [149] and [110]). Later, however, we discovered the following Lemma 8.24, which is more complicated but is more convenient for studying zeta-functions of root systems of higher ranks. Lemma 8.24 is a generalization of Lemma 8.13, and is proved inductively by using Lemma 8.7 (ii). We may find that the roles of Lemmas 8.7 (ii), 8.9 and 8.13 can be replaced by this single Lemma 8.24. This lemma has been essentially used in later developments of the u-method (such as [103] [110] [114] [116] and [117]). In this section, using Lemma 8.24 and Lemma 8.7 (i), we prove functional relations in the cases of type C2 . Lemma 8.24 ([103, Lemma 6.2]). For h ∈ N, let C = {C(l) ∈ C | l ∈ Z, l ̸= 0} ,

D = {D(N ; m; η) ∈ R | N, m, η ∈ Z, N ̸= 0, m ≥ 0, 1 ≤ η ≤ h} , A = {aη ∈ N | 1 ≤ η ≤ h}

be sets of numbers indexed by integers. Assume that (8.3.1) F (θ; C; D; A) =

X

(−1)N C(N )eN

√

−1θ

N ∈Z N ̸=0

− 2

aη h X X η=1 k=0

ϕ(aη − k)ϵaη +k

 ×

k  X X

 ξ=0  N ∈Z

(−1)N D(N ; k − ξ; η)eN

√

N ̸=0

−1θ

   (√−1θ)ξ  

ξ!

converges absolutely and is a constant function with respect to θ ∈ [−π, π]. Then, for d ∈ N0 , √ X (−1)N C(N )eN −1θ (8.3.2) Nd N ∈Z N ̸=0

−2

aη h X X η=1 k=0

ϕ(aη − k)ϵaη +k

 k−ξ  k X X ω+d−1 ξ=0

ω=0

ω

(−1)ω

8.3. FUNCTIONAL RELATIONS FOR THE ZETA-FUNCTION OF C2

X (−1)m D(m; k − ξ − ω; η)em × md+ω m∈Z

√

−1θ

139

 √ ( −1θ)ξ ξ!

m̸=0

+2

d X k=0

 η −1  k X h aX X ω+k−ξ

(−1)ω ω η=1 ω=0 ξ=0 √ X D(m; aη − 1 − ω; η)  ( −1θ)ξ × =0 mk−ξ+ω+1 ξ! m∈Z

ϕ(d − k)ϵd+k

m̸=0

holds for θ ∈ [−π, π], where the left-hand side converges absolutely and uniformly for θ ∈ [−π, π]. Proof. For d ∈ N0 , we set (8.3.3) Gd (θ; C; D; A) √ X aη h X X 1 (−1)l C(l)el −1θ = √ −2 ϕ(aη − k)ϵaη +k ld ( −1)d l∈Z η=1 k=0 l̸=0

×

ξ  k X X ξ=0

ν=0

 d−1+ξ−ν (−1)ξ ξ−ν

X (−1)m D(m; k − ξ; η)em × md+ξ−ν m∈Z

√

−1θ

√  (− −1θ)ν ν!

m̸=0

√ X aη h X X 1 (−1)l C(l)el −1θ = √ −2 ϕ(aη − k)ϵaη +k ld ( −1)d l∈Z η=1 k=0 l̸=0

×

k X k−ν  X ν=0

ω=0

 d−1+ω (−1)ω ω

X (−1)m D(m; k − ν − ω; η)em × md+ω m∈Z

√

−1θ

 √  ( −1θ)ν . ν!

m̸=0

It is noted that the second equality of (8.3.3) holds by setting ω = ξ−ν. Hence (8.3.1) means (8.3.4)

G0 (θ; C; D; A) = R0

(θ ∈ [−π, π])


for a constant R0 = R0 (C; D; A) ∈ C, by the relation −1+ξ−ν = 0 if ξ−ν ξ > ν. We can confirm that Gd (θ; C; D; A) converges absolutely with respect to θ ∈ [−π, π] and (8.3.5)

d Gd (θ; C; D; A) = Gd−1 (θ; C; D; A) (d ∈ N). dθ

140

8. FUNCTIONAL RELATIONS (I)

In fact, we differentiate term by term the second member of (8.3.3) in θ to obtain d Gd (θ; C; D; A) dθ √ X aη h X X 1 (−1)l C(l)el −1θ = √ −2 ϕ(aη − k)ϵaη +k ld−1 ( −1)d−1 l∈Z η=1 k=0 l̸=0

× −

ξ  k X X ν=0

ξ=0 ξ  X ν=1

√ √  X (−1)m D(m; k − ξ; η)em −1θ (− −1θ)ν d−1+ξ−ν ξ (−1) ξ−ν md+ξ−ν−1 ν! m∈Z m̸=0

X (−1)m D(m; k − ξ; η)em d−1+ξ−ν (−1)ξ ξ−ν md+ξ−ν m∈Z 

m̸=0

√

−1θ

√  (− −1θ)ν−1 . (ν − 1)!

We replace ν − 1 by µ in the last double sum and use the property       m−1 m m−1 − + = (l, m ∈ N). l−1 l l Then the right-hand side of the above can be written as √ X aη h X X 1 (−1)l C(l)el −1θ √ − 2 ϕ(aη − k)ϵaη +k ld−1 ( −1)d−1 l∈Z η=1 k=0 l̸=0

×

ξ  k X X ξ=0

ν=0

 d−2+ξ−ν (−1)ξ ξ−ν

X (−1)m D(m; k − ξ; η)em × md+ξ−ν−1 m∈Z

√

−1θ

√  (− −1θ)ν , ν!

m̸=0

which gives (8.3.5). Integrating both sides of (8.3.4) and multipling by √ −1 on both sides, we obtain √ √ −1G1 (θ; C; D; A) = R0 · −1θ + R1 for a certain constant R1 = R1 (C; D; A). If we repeat this operation and use (8.3.5), then (8.3.6)

√ ( −1θ)µ ( −1) Gd (θ; C; D; A) = Rd−µ µ! µ=0 √

d

d X

for certain constants Rµ = Rµ (C; D; A) (0 ≤ µ ≤ d). Now we aim to express {Rµ } explicitly. If we set θ = ±π in (8.3.6) with d + 1 (d ∈ N0 ),

8.3. FUNCTIONAL RELATIONS FOR THE ZETA-FUNCTION OF C2

141

then (8.3.7) √ √ [d/2] X ( −1)d+1 ( −1π)2ρ √ {Gd+1 (π; C; D; A) − Gd+1 (−π; C; D; A)} = Rd−2ρ . (2ρ + 1)! 2 −1π ρ=0 By (8.3.3) we see that the left-hand side can be expressed as (8.3.8) aη h X X −2 ϕ(aη − k)ϵaη +k η=1 k=0

×

[(k−1)/2]  k−2τ −1 

X

X

τ =0

ω=0

√  X D(m; k − 2τ − 1 − ω; η)  ( −1π)2τ d+ω ω (−1) , ω md+ω+1 (2τ + 1)! m∈Z m̸=0

where we put ν = 2τ + 1. Applying (8.2.6) with  x−1  X X D(m; x − 1 − ω; η) d+ω g(x) = (−1)ω , ω md+ω+1 m∈Z ω=0 m̸=0

we can express (8.3.7) as (8.3.9)  η −1  h aX X d+ω ω

η=1 ω=0

(−1)

ω

X D(m; aη − 1 − ω; η) md+ω+1

m∈Z m̸=0

√ ( −1π)2ρ = Rd−2ρ . (2ρ + 1)! ρ=0 [d/2]

X

From (8.3.9) we can deduce (8.3.10) Rµ = Rµ (C; D; A) = −2

µ X ν=0

ϕ(µ − ν)ϵµ+ν

 η −1  h aX X ν+ω η=1 ω=0

ω

(−1)ω

X D(m; aη − 1 − ω; η) m∈Z m̸=0

mν+ω+1

for µ ∈ N0 . In fact, putting d′ = d + 1 and g(x) = Rx−1 , we see that the right-hand side of (8.3.9) is equal to [(d′ −1)/2]

X ρ=0

g(d′ − 2ρ)

(−1)ρ π 2ρ . (2ρ + 1)!

Therefore the right-hand side of (8.3.10) is equal to −2

µ X ν=0

ϕ(µ − ν)ϵµ+ν

[(ν ′ −1)/2]

X ρ=0

g(ν ′ − 2ρ)

(−1)ρ π 2ρ (2ρ + 1)!

142

8. FUNCTIONAL RELATIONS (I) ′

= −2

µ X ν ′ =1

′

′

ϕ(µ − ν )ϵµ′ +ν ′

[(ν ′ −1)/2]

X ρ=0

g(ν ′ − 2ρ)

(−1)ρ π 2ρ , (2ρ + 1)!

where µ′ = µ + 1 and ν ′ = ν + 1. Using (8.2.6), we find that this is g(µ′ ) = Rµ , hence the claim follows. Combining (8.3.3),(8.3.6) and (8.3.10), we obtain (8.3.11) √ X (−1)l C(l)el −1θ ld

l∈Z l̸=0

×

−2

aη h X X η=1 k=0

 k X k−ν  X d−1+ω ν=0

ω

ω=0

ϕ(aη − k)ϵaη +k

(−1)ω

X (−1)m D(m; k − ν − ω; η)em × md+ω m∈Z

√

−1θ

 √ ( −1θ)ν ν!

m̸=0

= −2 ×

d−µ d X X µ=0 ν=0

ϕ(d − µ − ν)ϵd+µ+ν

 η −1  h aX X ν+ω η=1 ω=0

ω

√ X D(m; aη − 1 − ω; η) ( −1θ)µ . (−1) mν+ω+1 µ! m∈Z ω

m̸=0

On the right-hand side of (8.3.11), change the running index ν by k = µ + ν. Then the right-hand side of (8.3.11) can be written as  η −1  d k X h aX X X ω+k−µ −2 ϕ(d − k)ϵd+k (−1)ω ω µ=0 η=1 ω=0 k=0 √ X D(m; aη − 1 − ω; η) ( −1θ)µ × , mk−µ+ω+1 µ! m∈Z m̸=0

which implies (8.3.2).

□

Now we consider ζ2 (s1 , s2 , s3 , s4 ; C2 ) =

∞ X ∞ X

m−s1 n−s2 (m + n)−s3 (m + 2n)−s4 ,

m=1 n=1

defined by (3.3.23). Fix p ∈ N with p ≥ 2. It follows from (8.1.10), (8.1.11), (8.1.17) and (8.1.19) that lim Jp+1 (θ; p; u)

u→1

8.3. FUNCTIONAL RELATIONS FOR THE ZETA-FUNCTION OF C2

√

 p

( −1) X (−1)l el =  2 lp l∈Z

√

−1θ

−2

p X

l̸=0

j=0

ϕ(p − j)ϵp+j

143



√

j

( −1θ)  =0 j!

for θ ∈ (−π, π). Hence, for s ∈ R with s > 1 and x ∈ C with |x| = 1, we have   √ √ p l l −1θ j X X (−1) e ( −1θ)   (8.3.12) −2 ϕ(p − j)ϵp+j   p l j! l∈Z j=0 l̸=0

∞ X (−1)m xm em × ms m=1

√

−1θ

= 0,

which holds for θ ∈ [−π, π] because the left-hand side is continuous for θ ∈ [−π, π]. We stress that the principle of the u-method explained in Remark 8.11 is again working here; consider a vanishing quantity, and multiply it by another quantity to get a new non-trivial identity. From (8.3.12) we have √ X (−1)l+m xm e(l+m) −1θ (8.3.13) l p ms l∈Z, l̸=0 m≥1 l+m̸=0

−2

p X j=0

( ϕ(p − j) ϵp+j

= (−1)p+1

∞ X (−1)m xm em ms m=1

√

−1θ

) √ ( −1θ)j j!

∞ X xm ms+p m=1

for θ ∈ [−π, π]. We apply Lemma 8.24 with h = 1, a1 = p, d = q for q ∈ N and X xm C(N ) = (N ∈ Z, N ̸= 0), l p ms l̸=0, m≥1 l+m=N

( D(N ; µ; 1) =

xN Ns

0

(if µ = 0 and N ≥ 1), (otherwise).

Here we fix s and x, and regard the right-hand side of (8.3.13) as a constant. Therefore it follows from (8.3.2) that √ X (−1)l+m xm e(l+m) −1θ (8.3.14) lp ms (l + m)q l̸=0, m≥1 l+m̸=0

−2

p X j=0

ϕ(p − j) ϵp+j

 j  X q−1+j−ξ ξ=0

j−ξ

144

8. FUNCTIONAL RELATIONS (I) √ √ ∞ X (−1)m xm em −1θ ( −1θ)ξ × (−1) ms+q+j−ξ ξ! m=1  q j  X X p−1+j−ξ +2 ϕ(q − j) ϵq+j j−ξ j=0 ξ=0 √ ∞ X xm ( −1θ)ξ p−1 × (−1) =0 ms+p+j−ξ ξ! m=1 j−ξ

for θ ∈ [−π, π]. √ Here we replace x by −xe −1θ on the left-hand side of (8.3.14). Then √

X l∈Z, l̸=0 m≥1, l+m̸=0

−2

p X j=0

(−1)l xm e(l+2m) −1θ lp ms (l + m)q ϕ(p − j) ϵp+j

 j  X q−1+j−ξ

ξ=0 √ m 2m −1θ

q−1 √ ( −1θ)ξ ξ!

∞ X x e × (−1) ms+q+j−ξ m=1  q j  X X p−1+j−ξ +2 ϕ(q − j) ϵq+j 2p − 1 j=0 ξ=0 √ √ ∞ X (−1)m xm em −1θ ( −1θ)ξ p−1 × (−1) =0 ms+p+j−ξ ξ! m=1 j−ξ

for θ ∈ [−π, π]. As for the first term on the left-hand side, we separate the terms corresponding to the condition l + 2m = 0 and move them to the right-hand side. Then we can again apply Lemma 8.24, this time with (h, a1 , a2 , d) = (2, p, q, r) for r ∈ N, C(N ) =

X l̸=0, m≥1 l+m̸=0,l+2m=N

(

(−1)µ 0

q−1+µ q−1

(

(−1)p 0

p−1+µ p−1

D(N ; µ; 1) = D(N ; µ; 2) =

xm (N ∈ Z, N ̸= 0), lp ms (l + m)q





xN (N/2)−s−q−µ

xN N −s−p−µ

(if N ≥ 1 : even), (otherwise),

(if N ≥ 1), (otherwise).

8.3. FUNCTIONAL RELATIONS FOR THE ZETA-FUNCTION OF C2

145

We consequently obtain (8.3.15) √ X (−1)l xm e(l+2m) −1θ l∈Z, l̸=0 m≥1, l+m̸=0 l+2m̸=0

−2

p X j=0

lp ms (l + m)q (l + 2m)r

ϕ(p − j)ϵp+j

j j−ξ  X X ω+r−1

ω

(−1)

ξ=0 ω=0 √ ∞ X m 2m −1θ

ω



q−1+j −ξ −ω q−1



√ x e ( −1θ)ξ × (−1) 2r+ω m=1 ms+q+j−ξ+r ξ!   q j j−ξ  X X X ω+r−1 ω p−1+j −ξ −ω −2 ϕ(q − j)ϵq+j (−1) ω p−1 j=0 ξ=0 ω=0 √ √ ∞ X (−1)m xm em −1θ ( −1θ)ξ × (−1)p ms+p+j−ξ+r ξ! m=1   j p−1  r X X X ω + j − ξ ω p+q−2−ω +2 ϕ(r − j)ϵr+j (−1) ω q−1 j=0 ξ=0 ω=0 √ ∞ X 1 xm ( −1θ)ξ × (−1)p−1−ω j−ξ+ω+1 2 ms+q+j−ξ+p ξ! m=1   j q−1  r X X X ω + j − ξ ω p+q−2−ω +2 ϕ(r − j)ϵr+j (−1) ω p−1 j=0 ξ=0 ω=0 √ ∞ X xm ( −1θ)ξ × (−1)p =0 ms+p+j−ξ+q ξ! m=1 j−ξ−ω

1

for θ ∈ [−π, π]. Set (θ, x) = (π, 1) and consider the real part. First, we show: Claim 8.25. The first term on the left-hand side of (8.3.15) with (θ, x) = (π, 1) is equal to ζ2 (p, s, q, r; C2 ) + (−1)p ζ2 (p, q, s, r; C2 ) + (−1)p+q ζ2 (r, q, s, p; C2 ) + (−1)p+q+r ζ2 (r, s, q, p; C2 ). Proof. We divide the first term on the left-hand side into the cases l ≥ 1 and l ≤ −1. The part l ≥ 1 gives ζ2 (p, s, q, r; C2 ). As for the latter case, we replace l by −j and divide it into the cases j < m and j > m. The part j < m is, by putting m − j = n, equal to (−1)p ζ2 (p, q, s, r; C2 ). On the part j > m, we put j − m = n and we further divide into two parts according to m < n and m > n. Each

146

8. FUNCTIONAL RELATIONS (I)

of these two terms can also be written in terms of the zeta-function of C2 . □ As for the latter four terms on the left-hand side of (8.3.15) with θ = π, we apply (8.2.5) to the real part, which corresponds to the case that ξ is even. Since the zeta-function of C2 can be continued meromorphically to the whole space (see Corollary 7.5), we obtain the following. Theorem 8.26 ([103, Remark 8.1]). For p, q, r ∈ N with p ≥ 2,

ζ2 (p, s, q, r; C2 ) + (−1)p ζ2 (p, q, s, r; C2 ) + (−1)p+q ζ2 (r, q, s, p; C2 ) + (−1)p+q+r ζ2 (r, s, q, p; C2 )

= 2(−1)p × [p/2] p−2ℓ X X 1 p + q − 2ℓ − µ − 1r − 1 + µ ζ(2ℓ)ζ(p + q + r − 2ℓ + s) 2r+µ q−1 r−1 µ=0 ℓ=0 +

[q/2] X ℓ=0

q−2ℓ

ζ(2ℓ)ζ(p + q + r − 2ℓ + s)

[r/2]

+

X ℓ=0

+

[r/2] X ℓ=0

ζ(2ℓ)ζ(p + q + r − 2ℓ + s) ζ(2ℓ)ζ(p + q + r − 2ℓ + s)

X

(−1)

µ

µ=0 p−1 X µ=0 q−1 X µ=0



p + q − 2ℓ − µ − 1 p−1 

1 2r−2ℓ+µ+1 (−1)

µ+1



p+q−µ−2 q−1

p+q−µ−2 p−1







r−1+µ r−1



r − 2ℓ + µ r − 2ℓ

r − 2ℓ + µ r − 2ℓ





holds for all s ∈ C except for singularities of functions on both sides. Remark 8.27. Using a slightly more delicate discussion, we can show that this theorem holds for the case p = 1. Remark 8.28. Theorem 8.26 (with Remark 8.27) was essentially announced in [100, Theorem 2.1 and Example 2.1], though in [100] its proof was omitted and the statements are written in terms of the zeta-function of B2 . In [103], we gave the proof of the case when p, q, r are even integers in Theorem 8.26, and the general case was just stated. We give here the full proof of the above general form of Theorem 8.26. It is to be noted that the first published proof of Theorem 8.26 is due to Nakamura [163, Theorem 5.5]. His method is different from ours, a generalization of that in [162] (see Remark 8.4). In the same paper, Nakamura also gave a functional relation which connects ζ2 (s; C2 ) with ζ2 (s; A2 ).

8.3. FUNCTIONAL RELATIONS FOR THE ZETA-FUNCTION OF C2

147

Similar to Corollary 8.20, we obtain the following fact, from which examples listed at the end of Section 4.2 can now be immediately deduced. Corollary 8.29. For k, l, m, n ∈ N, (8.3.16)

ζ2 (2k, 2n, 2l, 2m; C2 ) + ζ2 (2k, 2l, 2n, 2m; C2 ) + ζ2 (2m, 2l, 2n, 2k; C2 ) + ζ2 (2m, 2n, 2l, 2k; C2 ) X k =2 ζ(2ν)ζ(2k + 2l + 2m − 2ν + 2n) ν=0

× +

2k−2ν X µ=0

l X ν=0

× +

µ=0

ν=0



2m − 1 + µ 2m − 1





2k + 2l − 2ν − µ − 1 2k − 1



2m − 1 + µ 2m − 1



ζ(2ν)ζ(2k + 2l + 2m − 2ν + 2n)

2k−1 X

m X

×

(−1)

µ

µ=0

ν=0

+

22m+µ

2k + 2l − 2ν − µ − 1 2l − 1

ζ(2ν)ζ(2k + 2l + 2m − 2ν + 2n)

2l−2ν X

m X

×



1



1 22m−2ν+µ+1

2k + 2l − µ − 2 2l − 1



2m − 2ν + µ 2m − 2ν



ζ(2ν)ζ(2k + 2l + 2m − 2ν + 2n)

2l−1 X

(−1)µ+1



µ=0

2k + 2l − µ − 2 p−1



2m − 2ν + µ 2m − 2ν

 .

In particular, for k, l ∈ N, (8.3.17) 2ζ2 (2k, 2l, 2l, 2k; C2 ) X k 2k−2ν X = ζ(2ν)ζ(4k + 4l − 2ν)

  2k + 2l − 2ν − µ − 1 2k − 1 + µ 22k+µ 2l − 1 2k − 1 ν=0 µ=0    l 2l−2ν X X 2k − 1 + µ µ 2k + 2l − 2ν − µ − 1 + ζ(2ν)ζ(4k + 4l − 2ν) (−1) 2k − 1 2k − 1 ν=0 µ=0    m 2k−1 X X 1 2k + 2l − µ − 2 2k − 2ν + µ + ζ(2ν)ζ(4k + 4l − 2ν) 22k−2ν+µ+1 2l − 1 2k − 2ν ν=0 µ=0 1



148

8. FUNCTIONAL RELATIONS (I)

+

m X ν=0

ζ(2ν)ζ(4k + 4l − 2ν)

2l−1 X

(−1)µ+1

µ=0



2k + 2l − µ − 2 2k − 1



2k − 2ν + µ 2k − 2ν

 .

From (8.3.17) we can immediately obtain the examples listed in Example 4.16. Example 8.30. Setting (p, q, r) = (2, 2, 2) and (2, 4, 2) in Theorem 8.26, we have (8.3.18)

ζ2 (2, s, 2, 2; C2 ) + ζ2 (2, 2, s, 2; C2 ) 39 3 = − ζ(s + 6) + ζ(2)ζ(s + 4), 16 2 (8.3.19) ζ2 (2, s, 4, 2; C2 ) + ζ2 (2, 4, s, 2; C2 ) 83 5 = − ζ(s + 8) + ζ(2)ζ(s + 6) + ζ(4)ζ(s + 4). 16 2 Setting s = 2 in (8.3.18), we recover the result given in Example 4.16: ζ2 (2, 2, 2, 2; C2 ) =

π8 . 302400

Setting s = 4 in (8.3.19), we obtain 53 π 12 . 6810804000 Setting (p, q, r) = (2, 1, 1) in Theorem 8.26, we have ζ2 (2, 4, 4, 2; C2 ) =

ζ2 (2, 1, s, 1; C2 ) + ζ2 (2, s, 1, 1; C2 ) + ζ2 (1, s, 1, 2; C2 ) − ζ2 (1, 1, s, 2; C2 ) 11 = ζ(2)ζ(s + 2) − ζ(s + 4). 8 In particular, setting s = 1, we have 1 11 (8.3.20) ζ2 (2, 1, 1, 1; C2 ) = ζ(2)ζ(3) − ζ(5). 2 16 Similar to Corollary 8.21, we can prove the following fact which includes (8.3.20). Proposition 8.31 ([200, Theorem]). For a, b, c, d ∈ N with 2 ∤ (a + b + c + d), (8.3.21)

ζ2 (a, b, c, d; C2 ) ∈ Q[π 2 , {ζ(2j + 1)}j∈N ].

Remark 8.32. Unlike Corollary 8.21, this proposition cannot be derived from Theorem 8.26. We can easily check, however, that several special cases come from Theorem 8.26, for example, ζ2 (2p, 2q + 1, 2q + 1, 2r + 1; C2 ) ∈ Q[π 2 , {ζ(2j + 1)}j∈N ], ζ2 (2p, 2q, 2q, 2r + 1; C2 ) ∈ Q[π 2 , {ζ(2j + 1)}j∈N ],

ζ2 (2p, 2q, 2r + 1, 2p; C2 ) ∈ Q[π 2 , {ζ(2j + 1)}j∈N ].

8.4. AN APPLICATION OF NAKAMURA’S METHOD

149

Example 8.33. As examples of Proposition 8.31 (see [200, Example in Section 3]), we can obtain 3 35 ζ2 (1, 1, 1, 2; C2 ) = − ζ(3)π 2 + ζ(5), 8 8 1 11 2 ζ2 (2, 1, 1, 1; C2 ) = ζ(3)π − ζ(5), 12 16 61 103 ζ2 (1, 2, 1, 3; C2 ) = ζ(5)π 2 − ζ(7), 48 8 43 73 ζ2 (2, 1, 2, 2; C2 ) = − ζ(5)π 2 + ζ(7), 48 8 5 71 2 ζ2 (2, 3, 1, 1; C2 ) = − ζ(5)π + ζ(7), 12 16 53 363 ζ2 (3, 1, 1, 2; C2 ) = − ζ(5)π 2 + ζ(7), 96 64 217 1437 ζ2 (1, 2, 3, 3; C2 ) = − ζ(7)π 2 + ζ(9), 48 32 265 1755 ζ2 (1, 3, 2, 3; C2 ) = − ζ(7)π 2 + ζ(9), 48 32 241 399 ζ2 (1, 3, 3, 2; C2 ) = − ζ(7)π 2 + ζ(9), 24 4 5 99 ζ2 (2, 2, 2, 3; C2 ) = ζ(7)π 2 − ζ(9), 16 32 235 1557 2 ζ2 (2, 3, 2, 2; C2 ) = − ζ(7)π + ζ(9), 48 32 5 99 ζ2 (2, 3, 3, 3; C2 ) = − ζ(9)π 2 + ζ(11), 16 32 991 6531 ζ2 (3, 2, 3, 3; C2 ) = − ζ(9)π 2 + ζ(11), 192 128 1021 6729 ζ2 (3, 3, 3, 2; C2 ) = − ζ(9)π 2 + ζ(11). 96 64

8.4. An application of Nakamura’s method to the zeta-function of G2 After the discovery of the u-method, some other methods for the proof of functional relations were introduced by several mathematicians. We already mentioned the method of Nakamura in Remark 8.4. Zhou, Bradley and Cai [222] proposed an alternative method of rather elementary nature, which was further developed by Ikeda and K. Matsuoka [73], who proved certain functional relations among zetafunctions of A2 , A3 and A4 (of the form that some of the variables are equal to 0). A method developed in the authors’ paper [120, Section

150

8. FUNCTIONAL RELATIONS (I)

6] for C2 is also of elementary nature. In the case of A2 , the papers [148] and [175] must also be mentioned. In this section we show the idea of Nakamura’s method, by describing a proof of a certain functional relation for the zeta-function of G2 . This is due to the authors’ paper [117]. Example 8.34. As stated in (3.3.28), the zeta-function of type G2 is defined by ζ2 (s; G2 ) = ζ2 (s1 , s2 , s3 , s4 , s5 , s6 ; G2 ) ∞ X ∞ X 1 . = s s s 1 2 3 m n (m + n) (m + 2n)s4 (m + 3n)s5 (2m + 3n)s6 m=1 n=1 Then the functional relation ([117, Example 4.1]) (8.4.1) ζ2 (s, 2, 1, 1, 1, 1; G2 ) + ζ2 (s, 1, 2, 1, 1, 1; G2 ) + ζ2 (1, 2, 1, 1, s, 1; G2 ) − ζ2 (1, 1, 2, 1, 1, s; G2 ) + ζ2 (1, 1, 1, 2, 1, s; G2 ) − ζ2 (1, 1, 1, 2, s, 1; G2 )   651 1 5 = ζ(2)ζ(s + 4) − − s+1 − ζ(s + 6) 8 2 2 · 3s+2 X cos(2πm/3) 9π X sin(2πm/3) + − 135 2 m≥1 ms+5 ms+6 m≥1   111 1 81 = ζ(2)ζ(s + 4) − − s+1 ζ(s + 6) + L(1, χ3 )L(s + 5, χ3 ) 8 2 4 holds for s ∈ C except for singularities on both sides, where L(s, χ3 ) is the Dirichlet L-function attached to the primitive odd Dirichlet character χ3 of conductor 3. The second equality of the above formula √ depends on the fact5 that L(1, χ3 ) = (π 3)/9. In particular, setting s = 1 in (8.4.1), we obtain that the left-hand side is equal to 2ζ2 (1, 2, 1, 1, 1, 1; G2 ). Hence we have (8.4.2)

ζ2 (1, 2, 1, 1, 1, 1; G2 ) 1 109 81 = ζ(2)ζ(5) − ζ(7) + L(1, χ3 )L(6, χ3 ). 2 16 8

√ is a special case of the well-known formula L(1, χ) = q −1 π −1τ (χ)B1,χ¯ for P any odd character χ mod q, where τ (χ) is the Gauss sum and B1,χ = q q −1 a=1 χ(a)a (see, e.g., [207, Chapter 4]). 5This

8.4. AN APPLICATION OF NAKAMURA’S METHOD

151

We give the proof6 of (8.4.1) by calculating X X 1 (8.4.3) S = s 2 m n (m + n)(m + 2n)(m + 3n)(2m + 3n) n̸=0 m≥1 m+n̸=0 m+2n̸=0 m+3n̸=0 2m+3n̸=0

in two ways. Claim 8.35. (8.4.3) is equal to the left-hand side of (8.4.1). The proof of this claim is similar to the case of C2 (see Claim 8.25), so we omit it. Therefore the remaining task is to show that (8.4.3) is equal to the right-hand side of (8.4.1). A direct calculation shows that (8.4.3) can be rewritten as (8.4.4) X X m≥1

n̸=0 l1 ̸=0 l2 ̸=0 l3 ̸=0 l4 ̸=0

1 ms n2 l 1 l 2 l 3 l 4

× e2π

√

1

Z

1

Z

0

0

1

Z 0

1

Z

e2π

√

√ −1x1 (m+n−l1 ) 2π −1x2 (m+2n−l2 )

√ −1x3 (m+3n−l3 ) 2π −1x4 (2m+3n−l4 )

e

e

0

dx1 dx2 dx3 dx4 .

It is possible to compute this value by using (4.2.4). This idea was initiated by Zagier (see [6, Proposition 4.14]) and systematically used by Nakamura (see [162] [163] and also [140] [141]). However, if we apply this method directly to (8.4.4), the necessary computations are enormous. Therefore, in order to reduce the total amount of computation, we modify (8.4.3) using partial fraction decompositions, before applying the idea of Zagier–Nakamura. By the partial fraction decomposition 1 1 1 = − , (m + 3n)(2m + 3n) m(m + 3n) m(2m + 3n) we divide (8.4.3) into two sums, X X∗ 1 (8.4.5) s+1 2 m n (m + n)(m + 2n)(m + 3n) X X∗ 1 − , s+1 2 m n (m + n)(m + 2n)(2m + 3n) P P∗ where denotes the same double sum as in (8.4.3). Then apply 1 1 1 = − , (m + 2n)(m + 3n) m(m + 3n) m(2m + 3n) 6There

are several misprints in the proof of (8.4.1) written in [117]. The following proof is the corrected version.

152

8. FUNCTIONAL RELATIONS (I)

1 2 1 = − (m + 2n)(2m + 3n) n(2m + 3n) n(m + 2n) to find that (8.4.5) is X X∗ X X∗ 1 1 2 − s+1 3 s+1 3 m n (m + n)(m + 2n) m n (m + n)(m + 3n) X X∗ 1 −2 . ms+1 n3 (m + n)(2m + 3n) Further, we use 1 1 1 = − , (m + n)(m + 3n) 2n(m + n) 2n(m + 3n) 1 1 2 = − . (m + n)(2m + 3n) n(m + n) n(2m + 3n) We obtain (8.4.6) 1 5 X X∗ 1 − ms+1 n3 (m + n)(m + 2n) 2 ms+1 n4 (m + n) X X∗ 1 X X∗ 1 1 + + 4 s+1 4 s+1 4 2 m n (m + 3n) m n (2m + 3n) 5 1 = 2Σ1 − Σ2 + Σ3 + 4Σ4 , 2 2

S=2

X X∗

say. We divide Σ1 as Σ1 =

X X∗∗

−

X X ∗∗ m+3n=0

−

X X ∗∗

,

2m+3n=0

P P∗∗ where denotes the sum over m ≥ 1, n ̸= 0, m + n ̸= 0 and m + 2n ̸= 0. Denote the first term by Σ11 . Putting m = 3l and n = −l in the second sum, we see that the second term is −

∞ X l=1

1 1 = ζ(s + 6). (3l)s+1 (−l)3 (3l − l)(3l − 2l) 2 · 3s+1

Similarly, putting m = 3l and n = −2l, we see that the third term is −3−s−1 2−3 ζ(s + 6). Therefore we obtain   1 1 1 − ζ(s + 6) = Σ11 + 3 s ζ(s + 6). (8.4.7) Σ1 = Σ11 + 2 · 3s+1 23 3s+1 23 As for Σ2 , we divide it as X X∗∗∗ X X ∗∗∗ X X ∗∗∗ X X ∗∗∗ (8.4.8) Σ2 = − − − , m+2n=0

m+3n=0

2m+3n=0

8.4. AN APPLICATION OF NAKAMURA’S METHOD

153

P P∗∗∗ where denotes the sum over m ≥ 1, n ̸= 0, m + n ̸= 0. Denote the first term by Σ21 and evaluate the remaining three sums as above to obtain   1 1 (8.4.9) Σ2 = Σ21 − + ζ(s + 6). 2s+1 24 3s−1 Similarly, we can obtain  (8.4.10)

Σ3 = Σ31 +

1 1 1 + s+1 + 4 s+2 2 2 23

 ζ(s + 6)

and  (8.4.11)

Σ4 = Σ41 + 1 −

1 2s+1

−



1 3s+2

ζ(s + 6),

where Σ31 =

m≥1

Σ41 =

1

X X n̸=0 m+3n̸=0

ms+1 n4 (m

m≥1

,

1

X X n̸=0 2m+3n̸=0

+ 3n)

ms+1 n4 (2m

+ 3n)

.

Consequently, we now arrive at the formula (8.4.12) 5 1 S = 2Σ11 − Σ21 + Σ31 + 4Σ41 + 2 2



17 1 5 − s+1 + 4 2 2 · 3s+2

 ζ(s + 6).

We evaluate the sums Σj1 (1 ≤ j ≤ 4). First, applying a partial fraction decomposition once more, we obtain X X∗∗ X X∗∗ 1 1 Σ11 = − . ms+1 n4 (m + n) ms+1 n4 (m + 2n) On the right-hand side, we separate the part m + 2n = 0 from the first double sum and separate the part m + n = 0 from the second double sum. We obtain   1 ♯ (8.4.13) Σ11 = Σ21 − Σ − 1 + s+1 ζ(s + 6), 2 where Σ♯ =

X X m≥1

n̸=0 m+2n̸=0

1 ms+1 n4 (m

+ 2n)

.

Next, we evaluate Σ21 . Recall the definition of the zeta-function of the root system A2 X 1 ζ2 (s1 , s2 , s3 ; A2 ) = . s s 1 2 m n (m + n)s3 m,n≥1

154

8. FUNCTIONAL RELATIONS (I)

The part corresponding to positive n of the sum Σ21 is exactly ζ2 (s + 1, 4, 1; A2 ). The part corresponding to negative n can be written as X 1 . s+1 n4 (m − n) m m,n≥1 m̸=n

By putting m − n = l when m > n and n − m = k when m < n, this is equal to X X 1 1 − s+1 4 s+1 (n + l) n l m,k≥1 m (m + k)4 k n,l≥1 = ζ2 (4, s + 1, 1; A2 ) − ζ2 (1, s + 1, 4; A2 ). Therefore (8.4.14) Σ21 = ζ2 (s + 1, 4, 1; A2 ) + ζ2 (4, s + 1, 1; A2 ) − ζ2 (1, s + 1, 4; A2 ) = −5ζ(s + 6) + 2ζ(2)ζ(s + 4) + 2ζ(4)ζ(s + 2),

where the second equality can be seen by Theorem 8.3. As for Σ♯ , we apply the idea of Zagier–Nakamura. Write Σ♯ as Z 1 √ X 1 ♯ (8.4.15) Σ = e2π −1(m+2n−l)θ dθ s+1 n4 l m 0 m≥1 n̸=0 l̸=0

1 = s+1 m m≥1 X

1

Z

e

√ 2π −1mθ

0

X e2π n̸=0

√

−1n·2θ

n4

X e2π l̸=0

√

−1l(−θ)

l

dθ,

where the interchange of the integration and summation is verified P because the sum l̸=0 is convergent uniformly in any closed subinterval of (0, 1) (see [148, Lemma 4.1]). Applying (4.2.4), we obtain √ Z 1 √ (2π −1)5 X 1 ♯ (8.4.16) Σ = e2π −1mθ B4 ({2θ})B1 ({−θ})dθ s+1 24 m 0 m≥1 √ (2π −1)5 X 1 = (J1 + J2 ), s+1 24 m m≥1 where 1/2

Z

e2π

J1 =

√

−1mθ

0

Z

1

e2π

J2 = 1/2

√

−1mθ

B4 (2θ)B1 (1 − θ)dθ,

B4 (2θ − 1)B1 (1 − θ)dθ.

Since B1 (x) = x − (1/2) and B4 (x) = x4 − 2x3 + x2 − (1/30), the factors B4 (2θ)B1 (1 − θ) and B4 (2θ − 1)B1 (1 − θ) are polynomials in θ of degree

8.4. AN APPLICATION OF NAKAMURA’S METHOD

155

5. It is easy to see recursively that (8.4.17) Z 1/2 √ k+1 X (−1)j−1+m k! (−1)k k! e2π −1mθ θk dθ = − (2πim)j 2k+1−j (k + 1 − j)! (2πim)k+1 0 j=1 and Z

1

(8.4.18)

e

√ 2π −1mθ k

1/2

θ dθ =

k+1 X j=1

(−1)j−1 k! (2πim)j (k + 1 − j)!



(−1)m 1 − k+1−j 2

 .

P Write B4 (2θ)B1 (1 − θ) = 5k=0 ak θk . (We see that a0 = −1/60, a1 = 1/30, a2 = 2, a3 = −12, a4 = 24 and a5 = −16.) Using (8.4.17), we find that the contribution of J1 to (8.4.16) is √ Z 1/2 √ 5 (2π −1)5 X 1 X a e2π −1mθ θk dθ k s+1 24 m 0 m≥1 k=0 ( √ 5 k+1 X (2π −1)5 X (−1)j−1 k! √ ak ϕ(s + j + 1) = 24 (2π −1)j 2k+1−j (k + 1 − j)! j=1 k=0  (−1)k k! √ − ζ(s + k + 2) . (2π −1)k+1 The contribution of J2 can be evaluated similarly. Substituting these expressions into (8.4.16) and using (8.1.1), we can obtain   π4 4π 2 1 ♯ (8.4.19) Σ = ζ(s + 2) + ζ(s + 4) − 16 + s ζ(s + 6), 45 3 2 because the terms ζ(s + k + 2) (k = 1, 3, 5) vanish. Substituting (8.4.14) and (8.4.19) into (8.4.13) we obtain   1 2 (8.4.20) Σ11 = −π ζ(s + 4) + 10 + s+1 ζ(s + 6), 2 and so 5 π4 17π 2 2Σ11 − Σ21 = − ζ(s + 2) − ζ(s + 4) 2 6  18 1 65 + + s ζ(s + 6). 2 2

(8.4.21)

The evaluation of Σ31 and Σ41 is similar to that of Σ♯ , but in these cases, we have to divide the integrals into three parts. As for Σ31 , we have Z 1 √ X 1 Σ31 = e2π −1(m+3n−l)θ dθ s+1 n4 l m 0 m≥1 n̸=0 l̸=0

156

8. FUNCTIONAL RELATIONS (I)

√ Z 1 √ (2π −1)5 X 1 = e2π −1mθ B4 ({3θ})B1 ({−θ})dθ s+1 24 m 0 m≥1 √ 5 X (2π −1) 1 = (J1′ + J2′ + J3′ ), s+1 24 m m≥1 where Ji′

Z

i/3

e2π

=

√

−1mθ

(i−1)/3

B4 (3θ − i + 1)B1 (1 − θ)dθ

(1 ≤ i ≤ 3).

We need, instead of (8.4.17) and (8.4.18), explicit expressions for Z i/3 √ Ii,k = e2π −1mθ θk dθ (1 ≤ i ≤ 3, k ∈ N0 ). (i−1)/3

Such expressions can be obtained recursively by using  √  √ 1 Ii,0 = √ e2π −1i/3 − e2π −1(i−1)/3 2π −1m and  k  k ! √ √ i i−1 1 2π −1i/3 2π −1(i−1)/3 Ii,k = √ e −e 3 3 2π −1m

k √ Ii,k−1 (k ≥ 1). 2π −1m Using those expressions we can evaluate J1′ , J2′ and J3′ , and the consequence is −

(8.4.22)

π4 ζ(s + 2) + 3π 2 ζ(s + 4) − 189ζ(s + 6) 45 X sin(2πm/3) X cos(2πm/3) + 18π − 216 . ms+5 ms+6 m≥1 m≥1

Σ31 =

The evaluation of Σ41 is similar. In this case we need the explicit expressions for Z i/3 √ e4π −1mθ θk dθ (1 ≤ i ≤ 3, k ∈ N0 ). (i−1)/3

The result is (8.4.23)

π4 3π 2 189 ζ(s + 2) + ζ(s + 4) − ζ(s + 6) 90 8 32 9π X sin(2πm/3) 27 X cos(2πm/3) − − . 8 m≥1 ms+5 4 m≥1 ms+6

Σ41 =

Therefore 1 π4 945 (8.4.24) Σ31 + 4Σ41 = ζ(s + 2) + 3π 2 ζ(s + 4) − ζ(s + 6) 2 18 8

8.4. AN APPLICATION OF NAKAMURA’S METHOD

+

157

X cos(2πm/3) 9π X sin(2πm/3) − 135 . s+5 s+6 2 m≥1 m m m≥1

Moreover, it is easy to see that X cos(2πm/3) 3−s−5 − 1 = ζ(s + 6), s+6 m 2 m≥1 √ √ X sin(2πm/3) 3 X χ3 (m) 3 = = L(s + 5, χ3 ). s+5 s+5 m 2 m 2 m≥1 m≥1 Therefore, from (8.4.12), (8.4.21) and (8.4.24), we can conclude that (8.4.3) coincides with the right-hand side of (8.4.1). This completes the proof of (8.4.1). Remark 8.36. Recall that the zeta-function of the root system C2 is defined by X 1 ζ2 (s1 , s2 , s3 , s4 ; C2 ) = . s s m 1 n 2 (m + n)s3 (m + 2n)s4 m,n≥1 Divide Σ11 into two subsums according to the conditions n ≥ 1 and n ≤ −1. Then the former part is exactly ζ2 (s+1, 3, 1, 1; C2 ). The latter is further divided according to the conditions m−n > 0 and m−n < 0. The part corresponding to m − n < 0 is −ζ2 (s + 1, 1, 3, 1; C2 ), while the remaining part is again divided into two subsums. The conclusion is that (8.4.25)

Σ11 = ζ2 (s + 1, 3, 1, 1; C2 ) − ζ2 (s + 1, 1, 3, 1; C2 )

+ ζ2 (1, 1, 3, s + 1; C2 ) − ζ2 (1, 3, 1, s + 1; C2 ).

On the other hand, we have shown that Σ11 can be written in terms of ζ(s) (see (8.4.20)). Combining these two formulas (8.4.25) and (8.4.20), we obtain a functional relation between the zeta-function of C2 and the Riemann zeta-function, which is different from the previously known relations ([103, Section 8] [163, Section 5]). Remark 8.37. We observed that the results in this chapter especially imply that some values of zeta-functions of root systems can be written in terms of the Riemann zeta values and/or L-values (see (8.2.38), (8.3.20), and (8.4.2)). The problem of expressing values of zeta-functions of root systems at integer points as a Q-linear combination of MZVs (of the same or lower depth) was raised in Zhao and Zhou [221], and such a result in the A3 case was given in the same paper. The case of the values of ζM T,r (s) was solved by Bradley and

158

8. FUNCTIONAL RELATIONS (I)

Zhou [21]. The case of zeta-functions of B2 and of G2 are treated, respectively, by Zhao [217] and [218], but in those cases, not only MZVs, but also special values of multiple polylogarithms (1.3.3) are necessary to state the results. Later, in the proof of Theorem 13.3 we will see that certain special values of zeta-functions of type Ar can be written in terms of MZVs. On the other hand, in the examples proved in this chapter, the values of zeta-functions of root systems are expressed by zeta (or L-) values of strictly lower depth. Those are examples of parity results, which will be more closely discussed in Section 12.5.

CHAPTER 9

Functional relations (II) In this chapter we develop another approach to functional relations for zeta-functions of root systems. As mentioned at the beginning of Chapter 8, this approach was first inspired by Nakamura’s alternative proof [162] of Theorem 8.3. In this approach, we can see the role of Weyl groups more explicitly. 9.1. A sketch in the case of A2 The fundamental idea in the present chapter is to introduce a certain sum, which is well behaved under the action of Weyl groups. In Chapter 4, to investigate special values at even integer points, we considered the sum S(s, y; ∆), which runs over the full lattice (see (4.1.3)). In this chapter, we will generalize this idea to introduce the sum S(s, y; I; ∆) (see (9.4.1)) which runs over a part of lattice, and has a nice Weyl-symmetric property. We will show that this sum yields functional relations among zeta-functions of root systems. Before going into the general formulation, we first explain briefly the essence of the idea in the simplest case of A2 . In this case, (4.1.3) is explicitly given as X 1 (9.1.1) S(s; A2 ) = s1 s2 m1 m2 (m1 + m2 )s3 m ,m ∈Z 1

2

m1 ,m2 ,m1 +m2 ̸=0

(refer to (3.3.19)). Now we define the following partial sum: X 1 (9.1.2) S(s, 0; {1}; A2 ) = . s1 s2 s3 m m (m 1 + m2 ) 1 2 m ∈N 1

m2 ∈Z m2 ,m1 +m2 ̸=0

e 2 , x3 ) we have Then by (4.1.11) and the definition of ϕ(s (9.1.3) S(s, 0; {1}; A2 ) X 1 X = s1 m1 m ∈N 1

m2 ∈Z\{0}

√ Z √ 1 −(2π −1)s3 1 e ϕ(s3 , x3 )e−2π −1(m1 +m2 )x3 dx3 s2 m2 Γ(s3 + 1) 0

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Komori et al., The Theory of Zeta-Functions of Root Systems, Springer Monographs in Mathematics, https://doi.org/10.1007/978-981-99-0910-0_9

159

160

9. FUNCTIONAL RELATIONS (II)

√ √ X 1 −(2π −1)s2 −(2π −1)s3 Z 1 √ e 3 , x3 )ϕ(s e 2 , x3 )e−2π −1m1 x3 dx3 . = ϕ(s s1 m Γ(s2 + 1) Γ(s3 + 1) 0 m ∈N 1 1

In particular, if s2 = k2 , s3 = k3 ∈ N, we have (9.1.4)

√ Z 1 √ (2π −1)k2 +k3 X 1 S(s, 0; {1}; A2 ) = Bk2 (x3 )Bk3 (x3 )e−2π −1m1 x3 dx3 . s1 k2 !k3 ! m1 0 m ∈N 1

For k ∈ N0 and m ∈ N, repeated application of integration by parts yields 1

Z (9.1.5) 0

xk e−2π

√

−1mx

dx =

k+1 X j=1

aj , (π −1m)j √

where aj ∈ Q. Therefore we obtain √  (2π −1)k2 +k3 X 1  1 √ (9.1.6) S(s, 0; {1}; A2 ) = P k2 !k3 ! ms11 π −1m m ∈N 1

√ = (2π −1)k2 +k3

k2 +k 3 +1 X j=1

b √ j j ζ(s1 + j), (π −1)

where P (x) ∈ Q[x] of degree k2 +k3 +1 and bj ∈ Q. Note that aj , bj can be explicitly determined. On the other hand, by dividing the region and changing the running indices (similar to Claims 8.25 and 8.35), S(s, 0; {1}; A2 ) is decomposed as (9.1.7)

S(s, 0; {1}; A2 ) = ζ2 (s1 , k2 , k3 ; A2 )

+ (−1)k2 ζ2 (k3 , k2 , s1 ; A2 ) + (−1)k2 +k3 ζ2 (k3 , s1 , k2 ; A2 ).

Therefore we conclude that a linear combination of ζ2 (s; A2 ) can be written in terms of those of zeta-functions of type A1 , that is, the Riemann zeta-function. In fact, by supplying the details of the above argument (see Subsection 9.8.2), we find that this result coincides with Theorem 8.3. In the following, we generalize the above method in arbitrary root systems; P (x) in (9.1.6) is a Bernoulli function (see Definition 9.9) and the linear combination in (9.1.7) is the orbit sum over a certain minimum coset representative (see Theorem 9.7). This chapter and the next chapter are mainly based on [113] [119] and [120].

9.2. LEMMAS ON ROOT SYSTEMS AND WEYL GROUPS

161

9.2. Lemmas on root systems and Weyl groups In this section we introduce further notations on root systems and Weyl groups, and prepare several root-theoretic lemmas. Let Aut(∆) be the subgroup of GL(V ) which stabilizes ∆ (see [70, Section 12.2]). Then the Weyl group W is a normal subgroup of Aut(∆) and there exists a subgroup Ω ⊂ Aut(∆) such that (9.2.1)

Aut(∆) = Ω ⋉ W.

The subgroup Ω is isomorphic to the group Aut(Γ) of automorphisms of the Dynkin diagram Γ. For w ∈ Aut(∆), we set ∆w = ∆+ ∩ w−1 ∆−

(9.2.2)

(which is the same as (4.1.6) when w ∈ W ), and define the length function by ℓ(w) = #∆w (see [71, Section 1.6]). The subgroup Ω is characterized as w ∈ Ω if and only if ℓ(w) = 0. Note that w∆w = ∆− ∩ w∆+ = −∆w−1 and ℓ(w) = ℓ(w−1 ). For any subset A ⊂ ∆, let HA∨ be the set of all v ∈ V which satisfies ⟨α∨ , v⟩ = 0 for some α ∈ A. In particular, H∆∨ is the set of all walls of Weyl chambers, which coincides with H∆∨ defined by (4.1.2). Let I ⊂ {1, 2, . . . , r}, and ΨI = {αi | i ∈ I} ⊂ Ψ. Let VI be the subspace of V spanned by ΨI . Then ∆I = ∆ ∩ VI is the root system in VI whose fundamental system is ΨI . For ∆I , we denote the corresponding coroot lattice, weight lattice, etc. by M M (9.2.3) Q∨I = Zαi∨ , PI = Zλi , i∈I

Q∨I

i∈I

∨

etc. Let ι : → Q be the natural embedding, and ι∗ : P → PI the projection induced from ι; that is, for λ ∈ P , ι∗ (λ) is defined as a unique element of PI satisfying ⟨ι(q), λ⟩ = ⟨q, ι∗ (λ)⟩ for all q ∈ Q∨I . Explicitly, we have X (9.2.4) ι∗ (n1 λ1 + · · · + nr λr ) = n i λi . i∈I

We define (9.2.5)

CI = {v ∈ C | ⟨Ψ∨I c , v⟩ = 0,

⟨Ψ∨I , v⟩ > 0},

where I c is the complement of I. Then the dimension of the linear span of CI is #I and we have a disjoint union a (9.2.6) C= CI , I⊂{1,...,r}

162

9. FUNCTIONAL RELATIONS (II)

where C is the fundamental Weyl chamber defined by (4.1.1). The collection of all sets wCI for w ∈ W and I ⊂ {1, . . . , r} is called the Coxeter complex (see [71, Section 1.15]; it should be noted that we use here a slightly different notation), which partitions V and we have a decomposition a (9.2.7) P+ = PI++ , I⊂{1,...,r}

where (9.2.8)

PI++ = P+ ∩ CI .

In particular, P∅++ = {0} and P{1,...,r}++ = P++ . Let WI = W (∆I ) be the subgroup of W generated by all the reflections associated with the elements in ΨI , and (9.2.9)

W I = {w ∈ W | ∆∨I+ ⊂ w∆∨+ }.

Then we have the following key lemmas to among zeta-functions. Note that the statements case I = ∅ and hence we deal with I ̸= ∅ in also that W (∆) = W (∆∨ ) and hence ℓ(w) = #(∆∨+ ∩ w−1 ∆∨− ).

functional relations hold trivially in the their proofs. Note #(∆+ ∩ w−1 ∆− ) =

Lemma 9.1. ([113, Lemma 1]) The subset W I coincides with the minimal (right) coset representatives {w ∈ W | ℓ(σi w) > ℓ(w) for all i ∈ I} of WI . Proof. Let w ∈ W I . Then ∆∨I+ ⊂ w∆∨+ , which implies ∆∨I+ ∩ w∆∨− = ∅. In particular, αi∨ ̸∈ w∆∨− for i ∈ I, which yields ∆∨+ ∩w∆∨− = (∆∨+ \ {αi∨ }) ∩ w∆∨− . Therefore
(9.2.10) σi (∆∨+ ∩ w∆∨− ) = σi (∆∨+ \ {αi∨ }) ∩ w∆∨− = (∆∨+ \ {αi∨ }) ∩ σi w∆∨− ⊂ ∆∨+ ∩ σi w∆∨−

and ℓ(σi w) ≥ ℓ(w). Since ℓ(σi w) = ℓ(w) ± 1, we have ℓ(σi w) = ℓ(w) + 1 and w is a minimal coset representative. Assume that w ∈ W satisfies ℓ(σi w) > ℓ(w) for all i ∈ I. Then we have

(9.2.11) σi (∆∨+ ∩w∆∨− ) = (∆∨+ \{αi∨ })∩σi w∆∨− ∪ {−αi∨ }∩σi w∆∨− . Since #σi (∆∨+ ∩ w∆∨− ) = ℓ(w) and #((∆∨+ \ {αi∨ }) ∩ σi w∆∨− ) ≥ ℓ(σi w) − 1 = ℓ(w), we have #({−αi∨ } ∩ σi w∆∨− ) = 0. It implies that (9.2.12)

σi (∆∨+ ∩ w∆∨− ) ⊂ (∆∨+ \ {αi∨ }) ∩ σi w∆∨− ⊂ ∆∨+ ,

9.2. LEMMAS ON ROOT SYSTEMS AND WEYL GROUPS

163

that is, no element of ∆∨+ ∩ w∆∨− is sent to ∆∨− by σi for i ∈ I, and hence Ψ∨I ∩ w∆∨− = ∅. Since ⟨∆∨+ , ρ⟩ > 0 and ⟨∆∨− , ρ⟩ < 0, we see that α∨ ∈ w∆∨− if and only if ⟨α∨ , wρ⟩ < 0. Thus ⟨Ψ∨I , wρ⟩ > 0 and hence ⟨∆∨I+ , wρ⟩ > 0. It follows that ∆∨I+ ∩ w∆∨− = ∅ and w ∈ W I . □ From Lemma 9.1, or by the definition of WI and W I , it follows that (9.2.13)

WI W I = W.

Lemma 9.2. ([113, Lemma 2]) (9.2.14)

ι∗−1 (PI+ ) = PI+ ⊕ PI c =

[

wP+ .

w∈W I

Proof. The first equality is clear. We prove the second equality. Assume w ∈ W I . Then for λ ∈ P+ , we have (9.2.15)

⟨∆∨I+ , wλ⟩ = ⟨w−1 ∆∨I+ , λ⟩ ⊂ ⟨∆∨+ , λ⟩ ≥ 0.

Hence wP+ ⊂ ι∗−1 (PI+ ). Conversely, assume λ ∈ ι∗−1 (PI+ ). Since #∆∨ < ∞, it is possible to fix a sufficiently small constant c > 0 such that 0 < |⟨∆∨ , cρ⟩| < 1. Then we see that λ + cρ is regular (see [70, Section 10.1]), i.e., 0 ̸∈ ⟨∆∨ , λ + cρ⟩ and the signs of ⟨α∨ , λ⟩ and ⟨α∨ , λ + cρ⟩ coincide if e ∨+ = {α∨ ∈ ∆∨ | ⟨α∨ , λ + cρ⟩ > ⟨α∨ , λ⟩ ̸= 0, because ⟨∆∨ , λ⟩ ⊂ Z. Let ∆ ∨ e is a positive system and hence there exists an element 0}. Then ∆ + e ∨+ = w∆∨+ . Since λ ∈ ι∗−1 (PI+ ), we have ∆∨ ⊂ ∆ e ∨+ . w ∈ W such that ∆ I+ e ∨ , λ + cρ⟩ > 0 Hence ∆∨I+ ⊂ w∆∨+ , that is, w ∈ W I . Moreover, ⟨∆ + implies ⟨∆∨+ , w−1 (λ + cρ)⟩ > 0 and ⟨∆∨+ , w−1 λ⟩ ≥ 0 again due to the integrality. Therefore λ ∈ wP+ . □ Lemma 9.3. ([113, Lemma 3]) For λ ∈ ι∗−1 (PI+ ), an element w ∈ W I satisfying λ ∈ wP+ (whose existence is assured by Lemma 9.2) is unique if and only if λ ̸∈ H∆∨ \∆∨I .

Proof. Assume α∨ ∈ ∆∨ \ ∆∨I and λ ∈ ι∗−1 (PI+ ) ∩ Hα∨ . Let w ∈ W I satisfy λ ∈ wP+ . Then σα λ = λ ∈ wP+ and hence w−1 λ = σw−1 α w−1 λ ∈ P+ , which further implies w−1 α∨ ∈ ∆′∨ , where ∆′∨ is a coroot system orthogonal to w−1 λ whose fundamental system is given by Ψ′∨ = {αi∨ ∈ Ψ∨ | ⟨αi∨ , w−1 λ⟩ = 0} (see [71, Section 1.12]). If Ψ′∨ ⊂ w−1 ∆∨I , then W (Ψ′∨ )Ψ′∨ = ∆′∨ ⊂ w−1 ∆∨I , and hence w−1 α∨ ∈ w−1 ∆∨I , which contradicts the assumption α∨ ̸∈ ∆∨I . Therefore there exists a fundamental coroot αi∨ ∈ Ψ′∨ \ w−1 ∆∨I , which satisfies σi w−1 λ = w−1 λ ∈ P+ by construction. Since w ∈ W I , we have w−1 ∆∨I+ ⊂ ∆∨+ \{αi∨ }. Hence σi w−1 ∆∨I+ ⊂ ∆∨+ , because σi (∆∨+ \{αi∨ }) ⊂ ∆∨+ . Then putting w′ = wσi , we have W I ∋ w′ ̸= w such that λ ∈ wP+ ∩ w′ P+ .

164

9. FUNCTIONAL RELATIONS (II)

Conversely, assume that there exist w, w′ ∈ W I such that w ̸= w′ and λ ∈ wP+ ∩ w′ P+ . This implies that w−1 λ = w′−1 λ is on a wall of C and hence λ ∈ H∆∨ . Let ∆′′∨ = {α∨ ∈ ∆∨ | λ ∈ Hα∨ } be a coroot system orthogonal to λ so that λ ∈ H∆′′∨ . Assume ∆′′∨ ⊂ ∆∨I . Then by λ = ww′−1 λ, we have ww′−1 ∈ Wλ and hence ww′−1 ∈ W (∆∨I ), because Wλ = W (∆′′∨ ) ⊂ W (∆∨I ) by the assumption. Since id ̸= ww′−1 ∈ W (∆∨I ), there exists a coroot α∨ ∈ ∆∨I+ such that β ∨ = ww′−1 α∨ ∈ ∆∨I− . Then, since w−1 (ww′−1 )∆∨I+ ⊂ ∆∨+ and w−1 ∆∨I+ ⊂ ∆∨+ , we have w−1 β ∨ ∈ ∆∨+ from the first inclusion and w−1 (−β ∨ ) ∈ ∆∨+ from the second one, which leads to a contradiction. Therefore λ ∈ Hα∨ for some α ∈ ∆′′∨ \ ∆∨I . □ Here we give the structures of the Weyl groups of the classical root systems, that is, of types Ar , Br , Cr and Dr . W (Ar ) ≃ Sr+1 ,

(9.2.16)

W (Br ) ≃ Sr ⋉ (Z/2Z)r ,

(9.2.17)

W (Cr ) ≃ Sr ⋉ (Z/2Z)r ,

(9.2.18)

W (Dr ) ≃ Sr ⋉ (Z/2Z)r−1 ,

(9.2.19)

where Sr denotes the symmetric group of degree r. 9.3. Automorphisms on Dynkin diagrams For s = (sα )α∈∆+ ∈ C#∆+ , we define an action of w ∈ Aut(∆) by (9.3.1)

(gs)α = sg−1 α ,

where we set s−α = sα for −α ∈ ∆− . This action and the convention are extensions of those already appeared in the proof of (4.1.7). First, we give simple functional relations related to the group Aut(Γ). Since g ∈ Aut(Γ) ⊂ Aut(∆) is a permutation of {α1 , . . . , αr }, we have (9.3.2)

g∆+ = ∆+ ,

gP++ = P++ .

The following theorem and examples have not been published before. Theorem 9.4. For s ∈ S, y ∈ V and g ∈ Aut(Γ), we have ζr (g −1 s, g −1 y; ∆) = ζr (s, y; ∆)

(9.3.3) Proof. (9.3.4)

ζr (g −1 s, g −1 y; ∆) =

X λ∈P++

e2π

√

−1⟨g −1 y,λ⟩

Y α∈∆+

1 ⟨α∨ , λ⟩sgα

9.3. AUTOMORPHISMS ON DYNKIN DIAGRAMS

X

=

e2π

√

=

X

e2π

√

Y

1

gα∈g∆+

⟨gα∨ , gλ⟩sgα

−1⟨y,gλ⟩

gλ∈gP++ −1⟨y,λ⟩

λ∈P++

165

Y

1

α∈∆+

⟨α∨ , λ⟩sα

,

where in the last equality, we rewrite gα, gλ as α, λ respectively and use (9.3.2). □ This relation is easily understood by observing the series expressions of zeta-functions as follows. Example 9.5. In the case of the root system of type Ar , by Proposition 3.5, we see the symmetry (9.3.5)

←→

i

r+1−i

in indices, which implies the symmetry in the variables (9.3.6)

←→

sij

sr+1−j,r+1−i .

In particular, we studied the case r = 2 in Example 8.22. Example 9.6. Another interesting case is type D4 . By Proposition 3.9, we have the explicit form of the zeta-function of type D4 as (9.3.7) ζr (s; D4 ) =

∞ X m1 =1

···

−s− 12

× m1

∞ X

+

−s+ 34

+

(m1 + m2 + m4 )−s14 (m2 + m4 )−s24 m4

m4 =1 −

−

−s− 23

(m1 + m2 )−s13 (m1 + m2 + m3 )−s14 m2 −

−s− 34

× (m2 + m3 )−s24 m3

+

(m1 + 2m2 + m3 + m4 )−s12 +

+

× (m1 + m2 + m3 + m4 )−s13 (m2 + m3 + m4 )−s23 . From this expression, we see the S3 -symmetry with respect to permutations of m1 , m3 , m4 . In fact, this symmetry comes from that of the Dynkin diagram of type D4 given by Figure 9.3.1. α❝1

α❝2

α❝3 ❝

α4

Figure 9.3.1. For example, the permutation (3 4) acts on s as (9.3.8) s+ 14

←→

s− 14 ,

s+ 24

←→

s− 24 ,

s+ 34

←→

s− 34 .

166

9. FUNCTIONAL RELATIONS (II)

9.4. General functional relations For I ⊂ {1, 2, . . . , r}, we define X (9.4.1) S(s, y; I; ∆) =

e2π

√

−1⟨y,λ⟩

λ∈ι∗−1 (PI+ )\H∆∨

Y

1

α∈∆+

⟨α∨ , λ⟩sα

.

When I = ∅, we see that S(s, y; ∅; ∆) is nothing but S(s, y; ∆) defined by (4.1.3). In the next theorem (which first appeared in [113]) we show that S(s, y; I; ∆) converges on S × V , which is defined in (3.1.10), and is a linear combination of Lerch analogues of the zeta-functions of the root system ∆, which are defined in (3.1.7). Theorem 9.7. For s ∈ S and y ∈ V , S(s, y; I; ∆) is absolutely convergent and we have  X Y (9.4.2) S(s, y; I; ∆) = (−1)−sα ζr (w−1 s, w−1 y; ∆). w∈W I

α∈∆w−1

Proof. By Lemma 9.2 and the Weyl group-invariance of H∆∨ , we have  [  ∗−1 (9.4.3) ι (PI+ ) \ H∆∨ = wP+ \ H∆∨ w∈W I

=

[ w∈W I

=

[

w(P+ \ H∆∨ ) wP++ .

w∈W I

Noting Lemma 9.3, we have X X Y √ (9.4.4) S(s, y; I; ∆) = e2π −1⟨y,wλ⟩ α∈∆+

w∈W I λ∈P++

1 ⟨α∨ , wλ⟩sα

.

Further, (9.4.5) e2π

√

−1⟨y,wλ⟩

Y α∈∆+

1 ⟨α∨ , wλ⟩sα = e2π = e2π

√

√

−1⟨w−1 y,λ⟩

−1⟨w−1 y,λ⟩

Y

1

α∈∆+

⟨w−1 α∨ , λ⟩sα

Y

1

α∈w−1 ∆+

⟨α∨ , λ⟩swα

by rewriting α as wα. When (9.4.6)

α ∈ w−1 ∆+ ∩ ∆− = −(∆+ ∩ w−1 ∆− ) = −∆w ,

we further replace α by −α. Then we have

,

9.4. GENERAL FUNCTIONAL RELATIONS

(9.4.7)

X

e2π

√

Y

1

α∈∆+

⟨α∨ , wλ⟩sα

−1⟨y,wλ⟩

λ∈P++

=

Y

=

 Y

(−1)−swα

 X

α∈∆w

e2π

√

−1⟨w−1 y,λ⟩

λ∈P++

167

Y

1

α∈∆+

⟨α∨ , λ⟩swα



(−1)−sα ζr (w−1 s, w−1 y; ∆),

α∈∆w−1

where we have used the fact that w∆w = −∆w−1 . The absolute convergence of S(s, y; I; ∆) is shown by Lemma 3.3. □ Let ∆(I) = ∆+ \ ∆I+ and d = #I c . We may find VI = {γ1 , . . . , γd } ⊂ ∆(I) such that V = VI ∪ ΨI becomes a basis of V . Let VI = V (∆(I) ) be the set of all such bases. In particular, let V = V∅ be the set of all linearly independent subsets V = {β1 , . . . , βr } ⊂ ∆+ . L For V ∈ VI , the lattice L(V∨ ) = β∈V Zβ ∨ is a sublattice of Q∨ . ∨ ∨ ∨ ∨ V Let {µV γ }γ∈V be the dual basis of V = VI ∪ ΨI , namely ⟨γk , µγl ⟩ = ∨ V ∨ V δk,l , ⟨αi∨ , µV αj ⟩ = δi,j , and ⟨γk , µαi ⟩ = ⟨αi , µγk ⟩ = 0. Let pVI⊥ be the projection defined by X X ∨ ∨ µV (9.4.8) pVI⊥ (v) = v − µV γ ⟨γ , v⟩ = α ⟨α , v⟩, α∈ΨI

γ∈VI

for v ∈ V . (The second equality can be easily seen by expressing P P V v = k ak µ V γk + i bi µαi .) It should be noted that the projection pVI⊥ depends only on VI in the following sense. Lemma 9.8. ([119, Lemma 2.1]) For any linearly independent subset ΦI = {β1 , . . . , β#I } ⊂ ∆I+ and U = VI ∪ ΦI , we have X X ∨ ∨ (9.4.9) pVI⊥ (v) = v − µU µU γ ⟨γ , v⟩ = β ⟨β , v⟩. γ∈VI

β∈ΦI

∨ Proof. Put u = β∈ΦI µU β ⟨β , v⟩ and we show pVI⊥ (v) = u. It is enough to check that ⟨γ ∨ , pVI⊥ (v)⟩ = ⟨γ ∨ , u⟩ for all γ ∈ U. For γ ∈ VI , we have D E X ′∨ ⟨γ ∨ , pVI⊥ (v)⟩ = γ ∨ , v − µV ⟨γ , v⟩ = ⟨γ ∨ , v⟩ − ⟨γ ∨ , v⟩ = 0, ′ γ

P

D

⟨γ ∨ , u⟩ = γ ∨ , v −

γ ′ ∈VI

X γ ′ ∈VI

E ′∨ µU ⟨γ , v⟩ = ⟨γ ∨ , v⟩ − ⟨γ ∨ , v⟩ = 0, ′ γ

because VI ⊂ V, U. For β ∈ ΦI , we have E XD X ′∨ ∨ ⟨β ∨ , u⟩ = β ∨ , µU ⟨β ′∨ , v⟩⟨β ∨ , µU β ′ ⟨β , v⟩ = β ′ ⟩ = ⟨β , v⟩, β ′ ∈ΦI

β ′ ∈ΦI

168

9. FUNCTIONAL RELATIONS (II)

while by writing β ∨ =

P

⟨β ∨ , pVI⊥ (v)⟩ = =

α∈ΨI

X α∈ΨI

aα α∨ , we have

∨ ⟨β ∨ , µV α ⟩⟨α , v⟩ =

DX α∈ΨI

X α∈ΨI

aα ⟨α∨ , v⟩

E

aα α∨ , v = ⟨β ∨ , v⟩. □

Let V , R, ϕ0 and {·}V,β be given just before Theorem 4.12. Using these notions, we now define Bernoulli functions of the root system ∆ associated with I and their generating functions. Definition 9.9. Assume that ∆ is irreducible. If ∆ is of type A1 , (I) then fix ϕ0 = cα1∨ with c > 0. For tI = (tα )α∈∆(I) ∈ C#∆ and λ ∈ PI , let (9.4.10) F (tI , y, λ; I; ∆)  X Y  = V∈VI

γ∈∆(I) \VI



tγ −

P

tγ  √ − 2π −1⟨γ ∨ , pVI⊥ (λ)⟩

V ∨ β∈VI tβ ⟨γ , µβ ⟩

√ exp(2π −1⟨y + q, pVI⊥ (λ)⟩)

×

1 #(Q∨ /L(V∨ ))

×

Y tβ exp(tβ {y + q}V,β ) , e tβ − 1 β∈V

X q∈Q∨ /L(V∨ )

I

and define Bernoulli functions P (k, y, λ; I; ∆) of the root system ∆ associated with I by the expansion X Y t kα α (9.4.11) F (tI , y, λ; I; ∆) = P (k, y, λ; I; ∆) . k α! (I) (I) k∈N#∆ 0

α∈∆

In Chapter 16, we will show: Theorem 9.10. (i) F (tI , y, λ; I; ∆) is holomorphic in a neighborhood of the origin with respect to tI . (ii) If ∆ is not of type A1 , then F (tI , y, λ; I; ∆) is continuous in y and is independent of the choice of ϕ0 . A special case of the last assertion (the independence of the choice of ϕ0 ) was already mentioned in Remark 4.13. The irreducibility is not essential either (see Remark 4.14).

9.4. GENERAL FUNCTIONAL RELATIONS

169

The fundamental formula in our theory is the following theorem, which is the first main result of this chapter. The proof of this theorem will also be given in Chapter 16. Theorem 9.11. ([119, Theorem 2.3]) Let sα = kα ∈ N for α ∈ ∆ and sα ∈ C for α ∈ ∆I+ . We assume that s ∈ S (see (3.1.10)). Then we have (I)

(9.4.12)

S(s, y; I; ∆)

√  Y  (2π −1)kα = (−1) kα ! α∈∆(I)   X Y 1 × P (k, y, λ; I; ∆). ⟨α∨ , λ⟩sα α∈∆ λ∈P #∆(I)

I+

I++

In the case I = ∅, clearly ΨI = ∅, VI = V, VI = V , ∆(I) = ∆+ , PI = {0}, and hence the only possible λ is λ = 0. Also in this case we write t = t∅ = (tα )α∈∆+ . Then (9.4.10), (9.4.11) and (9.4.12) are reduced to  X Y t P γ (9.4.13) F (t, y, 0; ∅; ∆) = tγ − β∈V tβ ⟨γ ∨ , µV β ⟩ V∈V γ∈∆+ \V X  Y tβ exp(tβ {y + q}V,β )  1 × #(Q∨ /L(V∨ )) e tβ − 1 ∨ ∨ β∈V q∈Q /L(V )

=

X #∆+

P (k, y, 0; ∅; ∆)

k∈N0

Y t kα α , kα ! α∈∆ +

and (9.4.14)

S(k, y; ∆) = S(k, y; ∅; ∆) √ Y  (2π −1)kα #∆+ = (−1) P (k, y, 0; ∅; ∆) kα ! α∈∆ +

for k = (kα )α∈∆+ ∈ K = N show:

#∆+

∩ S and y ∈ V . In Chapter 16 we will

Theorem 9.12. The above F (t, y, 0; ∅; ∆) coincides with F (t, y; ∆) defined by (4.2.11) in Section 4.2. This theorem with (9.4.13) clearly implies Theorem 4.12. Comparing the coefficients of expansions (4.2.11) and (9.4.13), we find (9.4.15)

#(∆+ )

P (k, y, 0; ∅; ∆) = P (k, y; ∆) (k ∈ N0

),

170

9. FUNCTIONAL RELATIONS (II)

where the right-hand side is defined by (4.2.2) in Section 4.2. (We note that (9.4.15) for k ∈ K0 also follows from (4.2.5) and (9.4.14).) Remark 9.13. Here we mention some history of our research. The formula (9.4.14) was already stated in [113, (123)] and [105, (3.10)]. In those papers, P (k, y; ∆) was defined by the multiple integral (4.2.2) and F (t, y; ∆) was simply defined as the generating function of P (k, y; ∆) (see (4.2.11)). However, in [105, Theorem 4.1] an alternative expression of F (t, y; ∆) (Theorem 4.12) was shown, which coincides with the special case I = ∅ of (9.4.10). The advantage of this form was discussed at the end of Chapter 5. Since this form has turned out to be fundamental, later in [119], we decided to use this form as the definition of F (t, y; ∆). In fact, in this way we defined a more general quantity F (tI , y, λ; I; ∆), which is (9.4.10). Hereafter the whole theory will be constructed on this definition. For example, P (k, y, λ; I; ∆) is now defined as the coefficients of the Taylor expansion of F (tI , y, λ; I; ∆). As mentioned above, the case I = ∅ of Theorem 9.11 was given in [105] [113]. The general case, however, becomes much more complicated. To establish Theorem 9.11 for general I, it is necessary to develop a detailed theory on lattice sums of hyperplane arrangements [115] (see Remark 16.47). The fundamental philosophy of the method in the present chapter is to combine two expressions of S(s, y; I; ∆), that is (9.4.2) and (9.4.12). It will produce the relation of the form (9.4.16) (A signed sum of zeta-functions of root systems) = (A series involving Bernoulli functions of root systems). If the left-hand side of (9.4.16) does not vanish identically, then this relation gives a non-trivial functional relation among zeta-functions of root systems. In the following sections we will show explicit computations on the right-hand side of (9.4.16), and will state several examples. On the other hand, concerning the left-hand side, it is important to know when it does not vanish. A criterion of the non-vanishing in terms of Poincar´e polynomials will be given in Chapter 10. Next, we state the second main result in the present chapter. We mentioned above that the situation for general I is much more complicated than the case I = ∅. However, the next theorem, in a sense,

9.5. AN EXTENSION

171

asserts the contrary: The generating function F (tI , y, λ; I; ∆) for general I can be deduced, in the following sense, from the case I = ∅. That is, the generating function for I = ∅ knows “everything”. This “converse” theorem will also be proved in Chapter 16. Theorem 9.14. ([119, Theorem 2.4]) Let I ⊂ {1, . . . , r}. Fix an order of the elements of the set ∆I+ by numbering as β1 , . . . , βd(I) , where d(I) = #∆I+ . Then for λ ∈ PI++ , we have (9.4.17) F (tI , y, λ; I; ∆) =

Res √

∨ ,λ⟩ tβd(I) =2π −1⟨βd(I)

···

 Y 1 F (t, y; ∆), tα −1⟨β1∨ ,λ⟩ α∈∆

Res √

tβ1 =2π

I+

which does not depend on the choice of the order of iterated residues. √ Remark 9.15. The point (2π −1⟨α∨ , λ⟩)α∈∆I+ is on some sinQ gular hyperplane of ( α∈∆I+ 1/tα )F (t, y; ∆). So when we calculate √ the residue at tβj = 2π −1⟨βj∨ , λ⟩, the remaining variables (tβj+1 , . . . , tβd(I) ) are to be located outside any singular hyperplanes. We postpone the proof of the above four theorems to Chapter 16, because it requires quite a lot of pages. In the remaining part of this chapter we discuss several consequences of those theorems. 9.5. An extension In the previous section we proved (9.4.14), which is valid for k = (kα )α∈∆+ ∈ K. In this section we present certain extended formulas, which are still valid even if some of kα ’s are zero (hence k is outside of S, because of (3.1.10)). Those formulas are necessary in Chapter 12. The fundamental idea is to consider the action of a certain differential operator to (9.4.13). For v ∈ V , and a differentiable function f on V , let (9.5.1)

(∂v f )(y) = lim

For α ∈ ∆+ , define

h→0

f (y + hv) − f (y) . h

∂ Dα = ∂ α∨ . ∂tα tα =0

Let ∆∗ ⊂ ∆+ be a root set (defined in Section 6.1), and let A = ∆+ \ ∆∗ = {ν1 , . . . , νN } ⊂ ∆+ , and define D A = D νN · · · D ν1 .

172

9. FUNCTIONAL RELATIONS (II)

Similarly, we define 1 ∂ 2 Dα,2 = ∂2∨ , 2 ∂t2α tα =0 α DA,2 = DνN ,2 · · · Dν1 ,2 . Further, let Aj = {ν1 , . . . , νj } (1 ≤ j ≤ N − 1), A0 = ∅, and VA = {V ∈ V | νj+1 ∈ / LhR [V ∩ Aj ] (0 ≤ j ≤ N − 1)}, where LhR [ · ] denotes the R-linear hull (linear span). Let R be as before, and let [ (9.5.2) HR = (HR∨ + q). R∈R q∈Q∨

Remark 9.16. It should be noted that y ∈ HR if and only if ⟨y + ∈ Z for some V ∈ V , β ∈ V, q ∈ Q∨ . In fact, if y ∈ HR then P ∨ we can write y = r−1 j=1 aj βj + q (aj ∈ R). We can find an element βr ∈ ∆ such that V = {β1 , . . . , βr } ∈ V . Then ⟨y − q, µV βr ⟩ = 0 ∈ Z. V Conversely, assume ⟨y + q, µβ ⟩ = c ∈ Z. Write V = {β1 , . . . , βr−1 , β}. Pr−1 ∨ ∨ Since this is a basis, we may write y + q = with j=1 aj βj + aβ V aj , a ∈ R. Then c = ⟨y + q, µβ ⟩ = a, especially a ∈ Z. Therefore aβ ∨ − q ∈ Q∨ , which implies y ∈ HR . q, µV β ⟩

Definition 9.17. For ∆+ \ ∆∗ = A = {ν1 , . . . , νN } ⊂ ∆+ , t∆∗ = {tα }α∈∆∗ and y ∈ V , we define (9.5.3)

F∆∗ (t∆∗ , y; ∆) =

X

(−1)#(A\V)

V∈VA

× ×



tγ

Y γ∈∆+ \(V∪A)

tγ −

1 ∨ #(Q /L(V∨ ))

P

β∈V\A tβ ⟨γ

X

 ∨ , µV ⟩ β

 Y t exp(t {y+q} )  β β V,β . tβ −1 e ∨

q∈Q∨ /L(V ) β∈V\A

We consider the action of DA to F to obtain the following theorem. Theorem 9.18. ([116, Theorem 3.7]) Let ∆+ \ ∆∗ = A = {ν1 , . . . , νN } ⊂ ∆+ and t∆∗ = {tα }α∈∆∗ . (i) For y ∈ V \ HR , we have

(9.5.4) DA F (t∆∗ , y; ∆) = DA,2 F (t∆∗ , y; ∆) = F∆∗ (t∆∗ , y; ∆), and hence is independent of choice of the order of A.

9.5. AN EXTENSION

173

(ii) For y ∈ V , the function F∆∗ (t∆∗ , y; ∆) is the continuous exten

sion of DA F (t∆∗ , y; ∆) in y in the sense that DA F (t∆∗ , y+cϕ0 ; ∆) tends continuously to F∆∗ (t∆∗ , y; ∆) when c → 0+, and is holomorphic with respect to t∆∗ around the origin. To show this theorem, we first prepare the following: Lemma 9.19. ([116, Lemma 9.1]) For B ⊂ ∆+ and V ∈ V , we have (9.5.5)

LhR [V ∩ B] = {v ∈ V | ⟨v, µV β ⟩ = 0 for all β ∈ V \ B}.

Proof. Let v be an element of the right-hand side. We write P v = β∈V cβ β and have cβ = 0 for all β ∈ V \ B and hence X (9.5.6) v= cβ β ∈ LhR [V ∩ B]. β∈V∩B

The converse is shown similarly.

□

Proof of Theorem 9.18. For t = (tα )α∈∆+ ∈ T, y ∈ V , V ∈ V , B ⊂ ∆+ and q ∈ Q∨ /L(V∨ ), let (9.5.7)

F (t, y; V, B, q) = (−1)#(B\V)



tγ

Y γ∈∆+ \(V∪B)

×

tγ −

P

β∈V\B tβ ⟨γ

 ∨ , µV ⟩ β

 Y t exp(t {y + q} )  β β V,β , e tβ − 1 β∈V\B

so that (9.5.8)

F (t, y; ∆) =

X V∈V

1 ∨ #(Q /L(V∨ ))

X q∈Q∨ /L(V∨ )

F (t, y; V, ∅, q).

Assume y ∈ V \ HR , and let (9.5.9)

Fj = F (t, y; V, Aj , q).

We calculate Dνj+1 Fj . First, since y ∈ / HR , noting Remark 9.16 we find that  X  ∨ ∨ Fj = (9.5.10) ∂νj+1 tβ ⟨νj+1 , µV ⟩ Fj . β β∈V\Aj

∨ Consider the case νj+1 ∈ V. Then ⟨νj+1 , µV β ⟩ = δνj+1 ,β and X ∨ (9.5.11) tβ ⟨νj+1 , µV β ⟩ = tj+1 , β∈V\Aj

174

9. FUNCTIONAL RELATIONS (II)

∨ Fj = tj+1 Fj . where we write tνj+1 = tj+1 for brevity. Hence we have ∂νj+1 Therefore we obtain (9.5.12)   Y tγ P Dνj+1 Fj = (−1)#(Aj \V) tγ − β∈V\(Aj ∪{νj+1 }) tβ ⟨γ ∨ , µV β ⟩

γ∈∆+ \(V∪Aj )

×



Y β∈V\(Aj ∪{νj+1 })

tβ exp(tβ {y + q}V,β )  , e tβ − 1

which is equal to Fj+1 because ∆+ \(V ∪(Aj ∪{νj+1 })) = ∆+ \(V ∪Aj ) and #((Aj ∪ {νj+1 }) \ V) = #(Aj \ V). ∨ Next, consider the case νj+1 ∈ / V. If ⟨νj+1 , µV β ⟩ = 0 for all β ∈ V \ Aj , then  X  ∨ ∨ Fj = (9.5.13) ∂νj+1 tβ ⟨νj+1 , µV β ⟩ Fj = 0 β∈V\Aj

and hence Dνj+1 Fj = 0. Otherwise, since (9.5.14)   ∂ tj+1 1 P = −P , ∨ V ∨ V ∂tj+1 tj+1 =0 tj+1 − β∈V\Aj tβ ⟨νj+1 , µβ ⟩ β∈V\Aj tβ ⟨νj+1 , µβ ⟩ we have (9.5.15) Dν Fj = (−1)#(Aj \V)+1



tγ

Y γ∈∆+ \(V∪Aj ∪{νj+1 })

×

tγ −

P

β∈V\Aj tβ ⟨γ

 ∨ , µV ⟩ β

 Y t exp(t {y + q} )  β β V,β . e tβ − 1 β∈V\Aj

By noting V \ (Aj ∪ {νj+1 }) = V \ Aj and #((Aj ∪ {νj+1 }) \ V) = #(Aj \ V) + 1, we find that the right-hand side is equal to Fj+1 . We see that the condition ⟨νj+1 , µV β ⟩ = 0 for all β ∈ V \ Aj is equivalent to the condition νj+1 ∈ LhR [V ∩ Aj ]. Therefore the above results can be summarized as ( 0 (νj+1 ∈ LhR [V ∩ Aj ]), (9.5.16) Dνj+1 Fj = Fj+1 (νj+1 ∈ / LhR [V ∩ Aj ]). Hence ( (9.5.17)

DA F 0 =

0 FN

(V ∈ / VA ), (V ∈ VA ).

Similar to the above calculations, we see that DA,2 F0 gives the same result as (9.5.17). Thus, since F0 = F (t, y; V, ∅, q), from (9.5.8) we obtain (9.5.4).

9.5. AN EXTENSION

175

Now let y ∈ V , and we show (9.5.18)

lim {y + q + cϕ0 }V,β = {y + q}V,β .

c→0+

In fact, when y ∈ V \ HR this claim is clear, so we assume y ∈ HR . Then y+q +cϕ0 ∈ / HR for all sufficiently small c > 0. If ⟨y+q, µV / Z, β ⟩ ∈ then V lim {y + q + cϕ0 }V,β = lim {⟨y + q + cϕ0 , µV β ⟩} = {⟨y + q, µβ ⟩} = {y + q}V,β .

c→0+

c→0+

If ⟨y + q, µV β ⟩ ∈ Z, then lim {y + q + cϕ0 }V,β  V lim {⟨y + q + cϕ0 , µV  β ⟩} = lim {c⟨ϕ0 , µβ ⟩} = 0  c→0+ c→0+   (if ⟨ϕ0 , µV β ⟩ > 0), = V V lim (1 − {−⟨y + q + cϕ , µ ⟩}) = lim (1 − {−c⟨ϕ  0 0 , µβ ⟩}) = 1 β  c→0+ c→0+   (if ⟨ϕ0 , µV β ⟩ < 0).

c→0+

Therefore we obtain the claim (9.5.18), which yields the assertion on the continuity in y. Finally, since F (t, y; ∆) is holomorphic with respect to t around
the origin (Theorem 9.10), so is DA F (t∆∗ , y; ∆) with respect to t∆∗ . The proof of Theorem 9.18 is thus complete. □ Define P∆∗ (k∆∗ , y; ∆) by F∆∗ (t∆∗ , y; ∆) =

X #(∆∗ )

k∆∗ ∈N0

P∆∗ (k∆∗ , y; ∆)

Y t kα α . k α! α∈∆∗

When y = 0, we write P∆∗ (k∆∗ , 0; ∆) = B∆∗ (k∆∗ ; ∆). We obtain the following extension of (9.4.14). #∆

Theorem 9.20. ([116, Theorem 3.9]) Let k = (kα )α∈∆+ ∈ N0 + with kα > 1 − ϵα (α ∈ ∆∗ , where ϵα is defined by (3.1.9)) and kα = 0 (α ∈ ∆+ \ ∆∗ ), and let k∆∗ = (kα )α∈∆∗ . We have  X Y (9.5.19) (−1)kα ζr (w−1 k, w−1 y; ∆) w∈W

α∈∆+ ∩w∆−

= (−1)

#(∆+ )

Y  (2πi)kα P∆∗ (k∆∗ , y; ∆) kα ! α∈∆∗

provided that the series expressions of all the zeta-functions on the lefthand side absolutely converge.

176

9. FUNCTIONAL RELATIONS (II)

Proof. First, assume y ∈ V \HR . Let k′ = (kα′ )α∈∆+ with kα′ = kα (α ∈ ∆∗ ), kα′ = 2 (α ∈ ∆+ \ ∆∗ = A). Then clearly k′ ∈ K, and hence by (4.1.3), (4.1.7) and (9.4.14), we have X Y 1 (9.5.20) S(k′ , y; ∆) = e2πi⟨y,λ⟩ ′ ∨ , λ⟩kα ⟨α α∈∆+ λ∈P \H∆∨  X Y ′ = (−1)kα ζr (w−1 k′ , w−1 y; ∆) α∈∆+ ∩w∆−

w∈W

= (−1)

#(∆+ )

Y ′  (2πi)kα P (k , y; ∆) . kα′ ! α∈∆ ′

+

Q

2 α∈A ∂α∨

Applying to the above. From the first line we observe that 2 each ∂α∨ produces the factor (2πi⟨α∨ , λ⟩)2 . Hence the factor ζr (w−1 k′ , w−1 y; ∆) on the second line is transformed into (2πi)2#A ζr (w−1 k, w−1 y; ∆). Therefore we have  X Y (9.5.21) (2πi)2#A (−1)kα ζr (w−1 k, w−1 y; ∆) w∈W

α∈∆+ ∩w∆−

= (−1)

#(∆+ )

Y

∂α2 ∨



α∈A

Y ′  (2πi)kα P (k , y; ∆) . kα′ ! α∈∆ ′

+

Since Y Y ′  ′  Y (2πi)2  (2πi)kα (2πi)kα = , ′ ! ′ ! k k 2! α α ∗ α∈∆ α∈∆ α∈A +

we have (9.5.22)

X w∈W

 (−1)kα ζr (w−1 k, w−1 y; ∆)

Y α∈∆+ ∩w∆−

= (−1)

#(∆+ )

Y ′  Y 1  (2πi)kα 2 ′ ∂ ∨ P (k , y; ∆) . 2 α kα′ ! α∈A α∈∆∗

From (4.2.11) it follows that (9.5.23)  Y 1 ∂2  ∂α2 ∨ F (t, y; ∆) = 2 ∂t2α tα =0 α∈A

X m=(mα )α∈∆+ mα ∈N0 (α∈∆∗ ) mα =2(α∈A)

Y 1  Y tmα α ∂α2 ∨ P (m, y; ∆) . 2 m α! ∗ α∈A α∈∆

By Theorem 9.18, we see that the left-hand side of (9.5.23) is equal to X Y tmα α (9.5.24) F∆∗ (t∆∗ , y; ∆) = P∆∗ (m∆∗ , y; ∆) . mα ! #(∆∗ ) α∈∆∗ m∆∗ ∈N0

9.6. THE ACTION OF Aut(∆)

177

Comparing (9.5.23) with (9.5.24) we find that Y 1  (9.5.25) ∂α2 ∨ P (k′ , y; ∆) = P∆∗ (k∆∗ , y; ∆). 2 α∈A Therefore (9.5.22) implies the desired result when y ∈ V \ HR . By the continuity with respect to y, the result is also valid in the case when y ∈ HR . □ From Theorem 5.11 and (9.5.25), we have the rationality for P (k∆∗ , y; ∆) in the following sense. P #(∆ ) Theorem 9.21. For any k ∈ N0 + and y = ri=1 yi αi∨ ∈ V , we have P∆∗ (k∆∗ , y; ∆) ∈ LhQ [1, y1 , . . . , yr ] = LhQ [1, {y1 }, . . . , {yr }]. ∆∗

Next, assume that ∆ is not simply-laced. Then we have the disjoint union ∆ = ∆l ∪ ∆s , where ∆l is the set of all long roots and ∆s is the set of all short roots. The next theorem is for even integer points.1 Theorem 9.22. ([116, Theorem 3.10]) Let ∆1 = ∆l (resp. ∆s ), ∆2 = ∆s (resp. ∆l ), and ∆j+ = ∆j ∩ ∆+ (j = 1, 2). Then ∆j+ (j = 1, 2) is a root subset of ∆+ . For s1 = k1 = (kα )α∈∆1+ with kα = k ∈ 2N (for all α ∈ ∆1+ ) and ν ∈ P ∨ /Q∨ , we have  Y  (−1)#(∆+ ) (2πi)kα (9.5.26) ζr (k1 , ν; ∆1+ ) = P∆1+ (k1 , ν; ∆) . #W k α! α∈∆ 1+

Proof. Let s = k = (kα )α∈∆+ with kα = k ∈ 2N (α ∈ ∆1+ ) and kα = 0 (α ∈ ∆2+ ). Then obviously ζr (k1 , ν; ∆1+ ) = ζr (k, ν; ∆). Noting this fact, and applying Theorem 9.20 to ∆∗ = ∆l or ∆s , we obtain the assertion. □ 9.6. The action of Aut(∆) In this section we treat the situation I = ∅, and study the action of Aut(∆) on S(s, y; ∆), F (t, y; ∆) and P (k, y; ∆). Note that P \ H∆∨ is an Aut(∆)-invariant set, because H∆∨ is Aut(∆)-invariant. An action of Aut(∆) is naturally induced on any function f in s and y as follows: For w ∈ Aut(∆), (9.6.1) 1It

(wf )(s, y) = f (w−1 s, w−1 y).

is written in [116, p.49] that if there is an odd kα , then both sides of (9.5.19) vanish, but this sentence is wrong.

178

9. FUNCTIONAL RELATIONS (II)

Theorem 9.23. ([113, Theorem 9]) For s ∈ S and y ∈ V , and for w ∈ Aut(∆), we have  Y  (9.6.2) (wS)(s, y; ∆) = (−1)−sα S(s, y; ∆), α∈∆w−1

if sα ∈ Z for α ∈ ∆w−1 . Proof. From (9.4.1), we have X (9.6.3) (wS)(s, y; ∆) =

e2πi⟨w

−1 y,λ⟩

λ∈P \H∆∨

Y

1

α∈∆+

⟨α∨ , λ⟩swα

.

−1

Rewriting λ as w λ and noting that P \ H∆∨ is Aut(∆)-invariant, we have X Y 1 (9.6.4) (wS)(s, y; ∆) = e2πi⟨y,λ⟩ ∨ ⟨wα , λ⟩swα α∈∆ =

λ∈P \H∆∨

+

X

Y

e2πi⟨y,λ⟩

w−1 α∈∆+

λ∈P \H∆∨

=

 Y

1 ⟨α∨ , λ⟩sα

 (−1)−sα S(s, y; ∆).

α∈∆w−1

□ Thus we have: Theorem 9.24. ([113, Theorem 10]) For s ∈ S and y ∈ V , we have S(s, y; ∆) = 0 if there exists an element w ∈ Aut(∆) such that w−1 s = s, w−1 y ≡ y (mod Q∨ ), sα ∈ Z for α ∈ ∆w−1 , and X (9.6.5) sα ̸∈ 2Z. α∈∆w−1

Proof. If w−1 s = s and w−1 y ≡ y (mod Q∨ ), then (wS)(s, y; ∆) = S(s, y; ∆). Therefore by Theorem 9.23   Y  −sα (9.6.6) 1− (−1) S(s, y; ∆) = 0. α∈∆w−1

Then the assumption (9.6.5) implies S(s, y; ∆) = 0.

□

We put t−α = −tα and define an action of Aut(∆) by (9.6.7)

(wt)α = tw−1 α .

If ∆ is of type A1 , then F (t, y; A1 ) = tet{y} /(et − 1) in (4.2.12) is an even or, in other words, Aut(∆)-invariant function except for y ∈ Z. In the multiple cases, F (t, y; ∆) is revealed to be really an Aut(∆)-invariant function. To show it, we need some lemmas.

9.6. THE ACTION OF Aut(∆)

179

Lemma 9.25 ([113, Lemma 12]). We have (9.6.8)

HR =

[ w∈W

w

r [ j=1

! (HΨ∨ \{α∨j } + Z αj∨ ) .

The set {HR∨ +q | R ∈ R, q ∈ Q∨ } is locally finite, i.e., for any y ∈ V , there exists a neighborhood U (y) such that U (y) intersects finitely many of these hyperplanes. e ∨ = ∆∨ ∩ HR∨ is a coroot system Proof. Fix R ∈ R. Then ∆ e ∨ . Let µ be a nonzero vector normal to HR∨ . Then so that R∨ ⊂ ∆ there exists an element w ∈ W such that w−1 µ ∈ C. Put w−1 µ = Pr P ≥ 0. Then α∨ = rj=1 aj αj∨ ∈ ∆∨ orthogonal j=1 cj λj with each cj P to w−1 µ should satisfy rj=1 aj cj = 0. Since aj are all nonpositive or nonnegative, we have aj = 0 for j such that cj ̸= 0. Hence cj = 0 e ∨ ⊂ ∆∨ is orthogonal to w−1 µ except for only one j, because w−1 ∆ with codimension 1. That is, w−1 µ = cλj for some c > 0. Therefore e ∨ and HR∨ = Hw(Ψ∨ \{α∨ }) = w(Ψ∨ \ {αj∨ }) is a fundamental system of ∆ j Lr ∨ ∨ wHΨ∨ \{α∨j } . Moreover, Q = wQ = i=1 Z wαi∨ , which implies (9.6.9) HR∨ + Q∨ = wHΨ∨ \{α∨j } + Z wαj∨ , Lr ∨ ∨ since i=1,i̸=j Z αi ⊂ HΨ∨ \{αj } . This shows that HR is contained in the right-hand side of (9.6.8). The opposite inclusion is clear. The local finiteness follows from the expression (9.6.8) and #W < ∞. □ Lemma 9.26. For y ∈ V \ HR and q ∈ Q∨ , we have (9.6.10)

V {y + q}V,β = {⟨y + q, µV β ⟩} = 1 − {−⟨y + q, µβ ⟩}.

Proof. Put R = V \ {β} and write X (9.6.11) y + q = aβ β ∨ + aγ γ ∨ ∈ aβ β ∨ + H R ∨ . γ∈R

Since y ∈ / HR implies y + q ∈ / HR by (9.6.8), we have aβ ∈ / Z. Hence V V ⟨y + q, µβ ⟩ = aβ ∈ / Z and {y + q}V,β = {⟨y + q, µβ ⟩} = 1 − {−⟨y + V q, µβ ⟩} = {aβ }. □ Let D = {y ∈ V | 0 ≤ ⟨y, λi ⟩ ≤ 1 (1 ≤ i ≤ r)}. Due to the local finiteness shown in Lemma 9.25 and the fact ∂D ⊂ HR , we may write a (9.6.12) D \ HR = D(ν) , ν∈J

(ν)

where by D we denote each open connected component of D \ HR P and J is a set of indices. Let |k| = α∈∆+ kα . We denote Q [y] = Pr ∨ Q [y1 , . . . , yr ] with y = i=1 yi αi .

180

9. FUNCTIONAL RELATIONS (II)

Theorem 9.27. ([113, Theorem 13]) The function P (k, y; ∆) is (ν) analytically continued to a polynomial function Bk (y; ∆) ∈ Q [y] from each D(ν) to the whole space C ⊗ V with its total degree at most |k|. In [113], the proof of this result was rather long and complicated, because it was based on an integral expression of F (t, y; ∆) (Theorem 5.10). However, now we can use the explicit expression (4.3.5) (Theorem 4.12), and hence we can give the following simple proof. Proof. We see that in (4.3.5), the discontinuity in y appears only at y ∈ HR . Thus by Lemma 9.26, for each D(ν) , q ∈ Q∨ and β ∈ V, there exists an m = m(ν, q, β) ∈ Z for which (9.6.13)

{y + q}V,β = m + ⟨y + q, µV β ⟩

holds on y ∈ D(ν) . Thus from Theorem 5.11 and Remark 5.13 we can conclude that P (k, y; ∆) is a polynomial function with rational coefficients in y ∈ D(ν) whose total degree is at most |k|. □ (ν)

Remark 9.28. Polynomials Bk (y; ∆) may be called the Bernoulli polynomials of root systems. It is possible to prove several properties (ν) of Bk (y; ∆), analogous to well-known properties of classical Bernoulli polynomials (see [113, Theorems 15–18]). However, we do not discuss the details of them here, because, to develop our theory, Bernoulli functions of root systems P (k, y, λ; I; ∆) are more convenient than (ν) Bk (y; ∆). An action of Aut(∆) is naturally induced on any function f in t and y as follows: For w ∈ Aut(∆), (9.6.14)

(wf )(t, y) = f (w−1 t, w−1 y).

The following is shown in [113, Theorem 11] via the integral expression. Here we show it by using the explicit form (4.3.5). Theorem 9.29. Assume that ∆ is an irreducible root system. If ∆ is not of type A1 , then (9.6.15)

(wF )(t, y; ∆) = F (t, y; ∆) #∆+

for t ∈ C#∆+ and y ∈ V , and for w ∈ Aut(∆). Hence for k ∈ N0 and y ∈ V ,  Y  (9.6.16) (wP )(k, y; ∆) = (−1)−kα P (k, y; ∆). α∈∆w−1

9.6. THE ACTION OF Aut(∆)

181

Remark 9.30. If k is in the region S of absolute convergence with respect to s, the relation (9.4.14) and Theorem 9.23 immediately imply (9.6.16), while if k ̸∈ S, it should be proved independently. Proof. First, let y ∈ V \HR . Let ∆ be any irreducible root system. Then w−1 y + q ∈ V \ HR for any w ∈ Aut(∆) and q ∈ Q∨ /L(V∨ ) (because HR is Aut(∆)-invariant). Therefore, for w ∈ Aut(∆), by Lemma 9.26 we have (9.6.17)  X Y t P wγ (wF )(t, y; ∆) = twγ − β∈V twβ ⟨γ ∨ , µV β ⟩ V∈V γ∈∆+ \V

 Y twβ exp(twβ {⟨w−1 y + q, µV ⟩})  1 β twβ − 1 |Q∨ /L(V∨ )| e q∈Q∨ /L(V∨ ) β∈V  X Y tγ P = tγ − β∈wV tβ ⟨γ ∨ , wµV w−1 β ⟩ V∈V X

×

γ∈w(∆+ \V)

×

1 ∨ |Q /L(V∨ )|

 Y tβ exp(tβ {⟨y + wq, wµV−1 ⟩})  w β

X q∈Q∨ /L(V∨ )

β∈wV

e tβ − 1

.

For V ∈ V , we put

(9.6.18) V′ = (wV ∩ ∆+ ) ∪ (−(wV ∩ ∆− )) = (wV ∪ (−wV)) ∩ ∆+ .

Since V ∈ V , we see that V′ ∈ V and hence this correspondence gives a bijection V → V . For β ∈ wV, we write X ′ (9.6.19) wµV = aγ µ V −1 γ . w β γ∈V′

Then if γ ∈ wV ∩ ∆+ , (9.6.20)

−1 V aγ = ⟨γ, wµV w−1 β ⟩ = ⟨w γ, µw−1 β ⟩ = δw−1 β,w−1 γ = δβ,γ ,

and if γ ∈ −(wV ∩ ∆− ), (9.6.21) −1 V aγ = ⟨γ, wµV w−1 β ⟩ = −⟨w (−γ), µw−1 β ⟩ = −δw−1 β,w−1 (−γ) = −δβ,(−γ) . Thus ( (9.6.22)

wµV w−1 β

and hence (9.6.23) X tβ ⟨γ ∨ , wµV w−1 β ⟩ = β∈wV

=

X

′

µV β ′ −µV −β

(β ∈ wV ∩ ∆+ ) (β ∈ wV ∩ ∆− ),

tβ ⟨γ ∨ , wµV w−1 β ⟩ +

β∈wV∩∆+

X

tβ ⟨γ ∨ , wµV w−1 β ⟩

β∈wV∩∆−

182

9. FUNCTIONAL RELATIONS (II)

X

=

′

tβ ⟨γ ∨ , µV β ⟩+

β∈wV∩∆+

X

=

β∈V′

tβ ⟨γ

∨

X

′

(−t−β )⟨γ ∨ , −µV −β ⟩

β∈wV∩∆−

′ , µV β ⟩.

Therefore Y

(9.6.24)

γ∈w(∆+ \V)

Y

=

γ∈w(∆+ \V)

= =

tγ −

tγ −

P

−

P β∈V′

Y tγ γ∈w(∆+ \V)∩∆+

−

Y

=

tγ −

γ∈∆+ \V′

P

tγ V ∨ t β∈wV β ⟨γ , wµw−1 β ⟩

tγ V′ ∨ β∈V′ tβ ⟨γ , µβ ⟩

Y tγ γ∈w(∆+ \V)∩∆+

P

P β∈V′

tγ ′ tβ ⟨γ ∨ , µV β ⟩ tγ ′ tβ ⟨γ ∨ , µV β ⟩

(−t−γ ) P ′ (−t−γ )− tβ ⟨γ ∨ , µV β ⟩

Y

γ∈w(∆+ \V)∩∆−

β∈V′

Y tγ γ∈w(∆− \(−V))∩∆+

−

P β∈V′

tγ ′ tβ ⟨γ ∨ , µV β ⟩

tγ , V′ ∨ β∈V′ tβ ⟨γ , µβ ⟩

where to show the last equality, we used

(9.6.25) w(∆+ \ V) ∩ ∆+ ∪ w(∆− \ (−V)) ∩ ∆+
= ∆+ ∩ w (∆+ \ V) ∪ (∆− \ (−V))

= ∆+ ∩ w (∆+ \ (V ∪ (−V)) ∪ (∆− \ (V ∪ (−V))




= ∆+ \ (wV ∪ (−wV)) Also we have (9.6.26)

= ∆+ \ V ′ .

Y tβ exp(tβ {⟨y + wq, wµV w−1 β ⟩}) e tβ − 1

β∈wV

=

Y β∈wV∩∆+

=

Y β∈wV∩∆+

tβ exp(tβ {⟨y + wq, wµVw−1 β ⟩}) e tβ − 1

β∈wV∩∆−

=

β∈wV∩∆+

e tβ − 1

′ ′ tβ exp(tβ {⟨y + wq, µVβ ⟩}) Y −t−β exp(−t−β {⟨y + wq, −µV−β ⟩}) e tβ − 1 e−t−β − 1 β∈wV∩∆ −

′

Y

tβ exp(tβ {⟨y + wq, wµVw−1 β ⟩})

Y

tβ exp(tβ {⟨y + wq, µVβ ⟩}) e tβ − 1

′

Y β∈(−wV)∩∆+

Y tβ exp(tβ {⟨y + wq⟩}V′ ,β ) = , e tβ − 1 β∈V′

tβ exp(tβ (1 − {−⟨y + wq, µVβ ⟩})) e tβ − 1

9.7. EXPLICIT GENERATING FUNCTIONS (#I = (r − 1) CASE)

183

where on the last equality we again used Lemma 9.26. Since q runs over all Q∨ /L(V∨ ), we see that wq runs over all (9.6.27)

wQ∨ /wL(V∨ ) = Q∨ /L(wV∨ ) = Q∨ /L(V′∨ ).

Therefore we find (9.6.28) 1 ∨ |Q /L(V∨ )| =

X

 Y tβ exp(tβ {⟨y + wq, wµV−1 ⟩})  w β e tβ − 1

q∈Q∨ /L(V∨ ) β∈wV

1 ∨ |Q /L(V′∨ )|

X q∈Q∨ /L(V′∨ )

 Y t exp(t {⟨y + q⟩} ′ })  β β V ,β . tβ − 1 e β∈V′

Substituting (9.6.24) and (9.6.28) into (9.6.17), we obtain the assertion. Lastly, to treat the case y ∈ HR , we assume that ∆ is not of type A1 . Then we can apply the continuity of F (t, y; ∆) in y (Theorem 9.10) to obtain the result. The proof is complete. □ Remark 9.31. The above proof shows that, even in the case when ∆ is of type A1 , the assertions of Theorem 9.29 are valid for y ∈ V \HR (= V \ Q∨ in this case). 9.7. Explicit generating functions (#I = (r − 1) case) In the following sections of this chapter we deduce the explicit form of generating functions and functional relations from the fundamental formulas given in Section 9.4. Since some examples in the case I = ∅ were already presented in Section 4.2, here we treat the case I ̸= ∅. Actually, however, we only discuss the cases #I = r − 1 and #I = 1, because the other cases are much more complicated. We use the notation δ□ , where some condition is inserted in □, defined as δ□ = 1 if the condition is satisfied, and = 0 otherwise. We first introduce the transposes p∗V⊥ of the projections pVI⊥ (deI fined by (9.4.8)) by X ∨ (9.7.1) ⟨u, pVI⊥ (v)⟩ = ⟨u, v⟩ − ⟨u, µV β ⟩⟨β , v⟩ β∈VI

= ⟨u −

X β∈VI

∨ ⟨u, µV β ⟩β , v⟩

= ⟨p∗V⊥ (u), v⟩ I

for v ∈ V , that is (9.7.2)

p∗V⊥ (u) = u −

X

I

β∈VI

∨ ⟨u, µV β ⟩β .

184

9. FUNCTIONAL RELATIONS (II)

Let I ⊂ {1, . . . , r}. In this section we consider the situation #I c = 1, and put I c = {k}. Then ΨI = {αi }i∈I , and we see that ∆(I)∨ = {α∨ =

r X i=1

ai αi∨ ∈ ∆∨+ | ak = ⟨α∨ , λk ⟩ ̸= 0}.

Since #VI = 1 in the present case, we have (9.7.3)

VI = {V = {β} ∪ ΨI }β∈∆(I) .

For V = {β} ∪ ΨI ∈ VI and γ ∈ ∆(I) \ {β}, from (9.7.2) we have (9.7.4)

∨ p∗V⊥ (γ ∨ ) = γ ∨ − ⟨γ ∨ , µV β ⟩β . I

We put bi = bi (β) = ⟨β ∨ , λi ⟩ (1 ≤ i ≤ r) so that β∨ =

(9.7.5)

r X

bi αi∨ .

i=1

Then we find λk . bk P Write y = y1 α1∨ + · · · + yr αr∨ and λ = ri=1 mi λi ∈ PI . Then µV β =

(9.7.6)

i̸=k

(9.7.7)

λk ∨ ⟨β , λ⟩ bk r r  X X bi  = m i λi + − m i λk b i=1 i=1 k

pVI⊥ (λ) = λ −

=

i̸=k r X i=1 i̸=k

i̸=k

 bi  m i λi − λ k . bk

Note that Q∨ /L(V∨ ) = {ak αk∨ }0≤ak 0. Then for q = ak αk∨ ∈ Q∨ , n ⟨y + q, λ ⟩ o n y + a o k k k = . (9.7.9) {y + q}V,β = bk bk Substituting the above results into (9.4.10), we obtain the following form of the generating function (under the identification y = (yi )1≤i≤r and λ = (mi )1≤i(̸=k)≤r ): (9.7.10) F ((tβ )β∈∆(I) , (yi )1≤i≤r , (mi )1≤i(̸=k)≤r ; I; ∆)

9.7. EXPLICIT GENERATING FUNCTIONS (#I = (r − 1) CASE)

=

185

tγ √ t − − 2π −1⟨p∗V⊥ (γ ∨ ), λ⟩ γ (I) (I) β∈∆ γ∈∆ \{β} I  n y + a o k k r t exp tβ    β √ X 1 X bi bk × exp 2π −1 mi yi − (yk + ak ) , tβ − 1 bk 0≤a 0, we find that K(R min(2∥f⃗∥)−1 ) ⊂ UR (B). Therefore, choosing 1 c−1 , 1 = min √ ⃗∥ f ∈B 2 d∥f

(16.3.48) we obtain

√ WR ⊂ K( dR) = K(c1 R min(2∥f⃗∥)−1 ) ⊂ Uc1 R (B).

(16.3.49)

□ Lemma 16.19 ([115, Lemma 4.10]). Assume Λ = B = {f1 , . . . , fd } with B = {B}. Let c1 , c2 be as in Lemma 16.18. For µ > 0 and k = (kf )f ∈Λ ∈ Nd0 there exists K > 0 such that for all y ∈ V ′ \ S d f ∈Λ1 (HΛ\{f } + Z ) and all sufficiently large N ∈ N, (16.3.50) X

e2π

√

−1⟨y,v⟩′

Y f ∈Λ+

v∈Zd ∩(WN \Uc2 N (B)) f (v)̸=0 (f ∈Λ+ ) f (v)=0 (f ∈Λ0 )

1 f (v)kf

1

≤ KN − µ+1 (log N )d   X X 1 1 × 1+ (1−{⟨y+w, f⃗B ⟩′ })− µ+1 +{⟨y+w, f⃗B ⟩′ }− µ+1 . ⃗ f ∈Λ1 w∈Zd /LhZ [B]

Proof. For brevity, we put (16.3.51)

G(y, v) = e2π

√

−1⟨y,v⟩′

Y f ∈Λ+

1 . • ⃗ (⟨f, v⟩′ + f )kf

We rearrange {f1 , . . . , fl } = Λ+ and {fl+1 , . . . , fd } = Λ0 and decompose (16.3.52) ( ) l f (v) ̸= 0 (f ∈ Λ+ ), [ d v ∈ Z ∩ (WN \ Uc2 N (B)) = Xj (N ) f (v) = 0 (f ∈ Λ0 ) j=1 with (16.3.53)    Xj (N ) = v ∈ Zd  

 |ℜf1 (v)|, . . . , |ℜfj−1 (v)| ≤ c2 N, |ℜfj (v)| > c2 N,   f (v) = ̸ 0 (1 ≤ i ≤ l), fi (v) = 0 (l+1 ≤ i ≤ d), i   |vk | ≤ N (1 ≤ k ≤ d)

344

16. LATTICE SUMS OF HYPERPLANE ARRANGEMENTS

for 1 ≤ j ≤ l. Let A = t (f⃗)f ∈B with f⃗ regarded as column vectors. We rewrite the series in terms of u = Av. Let (16.3.54)  |u +ℜf• |, . . . , |u +ℜf • |≤ c N, |u +ℜf• | >c N,   1 j−1 j−1 2 j j 2  1    • • d Yj (N ) = u ∈ Z ui + fi ̸= 0 (1 ≤ i ≤ l), ui + fi = 0 (l + 1 ≤ i ≤ d),       |each entry of A−1 u| ≤ N

for 1 ≤ j ≤ l. In Yj (N ) for a fixed (u1 , . . . , uj−1 , uj+1 , . . . , ud ) with sufficiently large N ∈ N, we see that uj runs over all integers such • • that c2 N − ℜfj < uj < Hj+ and −Hj− < uj < −c2 N − ℜfj for some Hj± = Hj± (u1 , . . . , uj−1 , uj+1 , . . . , ud ) ≥ 0, where Hj± are determined by the intersection point of the half line {(u1 , . . . , uj−1 , x, uj+1 , . . . , ud ) | ± S x > 0} and the boundaries ∂WN = 1≤l≤d {Av | |vk | ≤ N (k ̸= l), vl = ±N } of WN . If there is no intersection point, then we put Hj± = 0 accordingly. See Figure 16.3.1 for these sets and parameters. u2

Y2 (N) Y1 (N) −H1−

H1+ Uc2 N

u1

∂WN

Uc1 N

Figure 16.3.1. From the proof of Lemma 16.16 we evaluate (16.3.55) X G(y, v) v∈Xj (N )

≤

1 ⃗ #(Zd /LhZ [B])

X w∈Zd /LhZ

√ l ⃗B ′ X Y e2π −1⟨y+w,fi ⟩ ui  . • ki (u + f ) i i i=1 ⃗ u∈Y (N ) j [B]

16.3. PROOF OF THEOREM 16.2

345

Further, for each j and w, we have (16.3.56)

X

l Y e2π

=

−1⟨y+w,f⃗iB ⟩′ ui •

(ui + fi )ki

i=1

u∈Yj (N )

√

!

X

l Y e2π

•

i=1

ui ∈Z, ui +fi ̸=0

(1≤i≤l)

•

|ui +ℜfi |≤c2 N

√

−1⟨y+w,f⃗iB ⟩′ ui •

(ui + fi )ki

!

(1≤i≤j−1)

•

c2 N −ℜfj 0, −π < ϕ < π and |ϕ + arg x| < π/2. Define (17.1.5)

g2 (s1 , s2 ) = ζEZ,2 (s1 , s2 ) −

Γ(1 − s1 ) Γ(s1 + s2 − 1)ζ(s1 + s2 − 1). Γ(s2 )

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Komori et al., The Theory of Zeta-Functions of Root Systems, Springer Monographs in Mathematics, https://doi.org/10.1007/978-981-99-0910-0_17

389

390

17. MISCELLANEOUS RESULTS

In [135], the formula (17.1.6) g2 (s1 , s2 ) = (2π)s1 +s2 −1 Γ(1 − s1 ) n √ o √ × eπ −1(s1 +s2 −1)/2 F+ (s1 , s2 ) + eπ −1(1−s1 −s2 )/2 F− (s1 , s2 ) and the symmetric property (17.1.7)

√ F± (1 − s2 , 1 − s1 ) = (±2π −1)s1 +s2 −1 F± (s1 , s2 )

for F± were shown (actually in a slightly generalized form). Combining (17.1.6) and (17.1.7), we readily obtain: Theorem 17.1. It holds that (17.1.8) g2 (s1 , s2 ) = Γ(1 − s1 ) {F+ (1 − s2 , 1 − s1 ) + F− (1 − s2 , 1 − s1 )} . This form is similar to (17.1.2). Moreover, since Ψ(s2 , s1 + s2 ; √ ±2π −1k) has the asymptotic expansion whose first term is √ (±2π −1k)−s2 , we may regard that F± (s1 , s2 ) is a kind of generalized Dirichlet series. From this viewpoint, we may understand that (17.1.8) is a functional equation for g2 , hence for ζEZ,2 . Theorem 17.1 is not explicitly stated in [135]; it first appeared in [138]. On the other hand, in [135], the formula (17.1.9) g2 (s1 , s2 ) g2 (1 − s2 , 1 − s1 ) √ = s +s −1 (2π) 1 2 Γ(1 − s1 ) ( −1)s1 +s2 −1 Γ(s2 ) π  √ + 2 −1 sin (s1 + s2 − 1) F+ (s1 , s2 ) 2 was shown. Later in [107] it was pointed out that the following theorem is immediate from (17.1.9): Theorem 17.2. The symmetric form g2 (s1 , s2 ) (17.1.10) = s 1 (2π) +s2 −1 Γ(1 − s1 )

of the functional equation g2 (1 − s2 , 1 − s1 ) √ ( −1)s1 +s2 −1 Γ(s2 )

is valid on the hyperplane {(s1 , s2 ) ∈ C2 | s1 + s2 = 2k + 1}, where k is any non-zero integer. Therefore we may interpret (17.1.9) as a “meta” functional equation which produces (17.1.10) for all k ̸= 0. A generalization of Theorem 17.2 to the case of double L-functions twisted by Dirichlet characters was given in [111]. As we discussed in Chapter 3, the Mordell–Tornheim multiple zetafunction ζM T,r is a specialization of the zeta-function of the root system

17.2. THE DESINGULARIZATION

391

of type Ar (see (3.2.6)). In this case, a functional equation can be shown not only for r = 2, but also for all r ≥ 2. This was done by Okamoto and Onozuka [169], who proved an analogue of (17.1.9) for ζM T,r , r ≥ 2. Another type of functional equation in the general r-fold case, due to Hirose and Sato [62], is also to be mentioned; this is valid for normalized Shintani multiple L-functions.

17.2. The desingularization In the case of the Riemann zeta-function ζ(s), the explicit values at negative integer points are important from various viewpoints, such as the p-adic theory. Therefore it is natural to study the values of multiple zeta-functions at negative integer points. However, as we remarked in Chapter 7 (see Remark 7.2), multi-variable multiple zeta-functions in general have many singular analytic sets. For example, ζEZ,r (s1 , . . . , sr ) has infinitely many singular hyperplanes (1.3.6). Almost all negative integer points are located on those singularities, and hence those points are the points of indeterminacy of ζEZ,r (s1 , . . . , sr ). This fact makes the matter of values of multiple zeta-functions at negative integer points very complicated. In fact, let k = (k1 , . . . , kr ) be one of such points. Then the “value” ζEZ,r (k) may be only defined as the limit (17.2.1)

lim ζEZ,r (s)

s→k

(s = (s1 , . . . , sr )),

but this value may depend on the choice of the way how s approaches k. The explicit evaluation of this value under various choices of the approaching way was first studied by Akiyama, Egami and Tanigawa [1], Akiyama and Tanigawa [3], and then by subsequent papers ([179] [180] [97] [98] [176] [40] [41] [144]). Among them, the first-named author’s article [98] treats a rather general class of multiple zeta-functions, involving zeta-functions of root systems. Therefore [98] includes the study of the values of zeta-functions of root systems at negative integer points in the above “limit” sense. On the other hand, there are also attempts to find some suitable way of regularization, which will give a more definite meaning of the “value” at k. Some people developed the method of renormalization in the frame of the Hopf algebra structure ([54] [129] [34] [35]). Another method is desingularization, due to Furusho and the authors ([44] [46]),

392

17. MISCELLANEOUS RESULTS

which is of a more analytic flavor.1 In [44] we only studied the case of (a slight generalization of) the Euler–Zagier multiple zeta-functions, but in [46], more generally, we treated multiple zeta-functions of the form ζr (s1 , . . . , sn ; A , b) defined by (7.2.25) (actually, moreover, with twisted factors on the numerators). The desingularized zeta-function associated with ζr (s1 , . . . , sn ; A , b) is defined as follows: (17.2.2) ζrdes (s1 , . . . , sn ; A × Qn ×

k=1

k=1

√ 2π −1sj

− 1)Γ(sj )

C n j=1

exp

n j=1

s −1 xj j

r X

exp

k=1

! ajk − bj

! xj

dxj



1 P

(1 − c)−1 n Y

Z

 

c→1

1

j=1 (e

r Y

, b) = lim

r Y



ajk xj − 1

−

c ,  P  n exp c j=1 ajk xj − 1

where the limit is taken for c ∈ R, and C is the Hankel contour, the same as defined in (8.1.4). This definition2 seems complicated, but we can show that the desingularized zeta-function ζrdes is actually a finite linear combination of ζr . Assume that there exists a set of constants cmj (1 ≤ m ≤ r, 1 ≤ j ≤ n) such that n X (17.2.3) cmj ajk = δm,k j=1

for all k, m, where δm,k is Kronecker’s delta. Let ( ! n !) r n Y X X −1 G(u, v) = 1− 1+ ckj (vj − bj ) ajk uj vj , k=1

j=1

j=1

where u = (uj ), v = (vj ) (j = 1, . . . , n) are indeterminates. Expanding the right-hand side, we may write G(u, v) =

X l,m

1The

αl,m

n Y

l

m

ujj vj j ,

j=1

principles of these two methods were originally quite different from each other, but actually these are closely connected; see Komiyama [96]. 2The original motivation for introducing this definition is connected with the theory of p-adic multiple L-functions; see [111] [45] and [46].

17.2. THE DESINGULARIZATION

393

where l = (l1 , . . . , ln ) ∈ Nn0 , m = (m1 , . . . , mn ) ∈ Zn , and αl,m are coefficients. The sum on the right-hand side is obviously a finite sum. Theorem 17.3. ([46, Theorem 2.7]) Under the assumption (17.2.3), we have (17.2.4) ζrdes (s1 , . . . , sn ; A , b) =

X

n Y

αl,m

! ζr (s1 + m1 , . . . , sn + mn ; A , b),

(sj )lj

j=1

l,m

where (s)k = s(s + 1) · · · (s + k − 1) stands for the Pochhammer symbol. Theorem 17.4. ([46, Theorem 2.2]) ζrdes (s1 , . . . , sn ; A , b) can be continued to the whole space Cn as an entire function. Theorem 17.4 justifies the name of the desingularized zeta-function. Theorem 17.3 implies that just a finite linear combination eliminates all of the infinitely many singularities. In particular, the values of ζrdes (s1 , . . . , sn ; A , b) at negative integer points are definitely determined, and we can write down those values explicitly: Theorem 17.5. ([46, Theorem 2.6]) Let λ1 , . . . , λn ∈ N0 . Then we have (17.2.5)

ζrdes (−λ1 , . . . , −λn ; A , b) =

n Y

λj

(−1) λj !

j=1

×

r Y

X mj +νj1 +···+νjr =λj (1≤j≤n)

P n Y ( r

k=1

j=1

ajk − bj )mj mj !

!

! Bν1k +···+νnk +1

k=1

ν n Y r Y ajkjk

ν ! j=1 k=1 jk

! .

Zeta-functions of root systems are special cases of ζr (s1 , . . . , sn ; A , b), with n = #(∆+ ), A = (⟨α∨ , λk ⟩)α∈∆+ ,1≤k≤r , and b = (⟨α∨ , ρ⟩)α∈∆+ . The assumption (17.2.3) is satisfied with (cmα ) = (Ir 0), where Ir is the r × r identity matrix, 0 is the (n − r) × r zero-matrix (provided that we fix the order in the set ∆+ so that the first r elements are fundamental roots). Then   r Y X 1 − G(u, v) = ⟨α∨ , λk ⟩uα vα vα−1  k

k=1

α∈∆+

(see [46, (3.12)]). By Theorems 17.4 and 17.5, we find that ζrdes (s; ∆) is entire, and its values at negative integer points can be explicitly computed.

394

17. MISCELLANEOUS RESULTS

Remark 17.6. Another variant of zeta-functions of root systems is their q-analogue, which was introduced and studied by Kato [88]. 17.3. The number of representations of SU (3) Let p1 (n) be the number (up to equivalence) of n-dimensional representations of sl(2), or equivalently, of SU (2). Since SU (2) has just one irreducible representation of dimension k for each k ∈ N, we see that p1 (n) is equal to the number of partitions of n. Therefore the asymptotic behavior of p1 (n) can be described by the classical Hardy– Ramanujan formula [55]: √ 1 (17.3.1) p1 (n) = √ eπ 2n/3 (1 + o(1)) (n → ∞). 4 3n Romik [177] considered the analogue of this matter in the case of sl(3), or SU (3). Let p2 (n) be the number of n-dimensional representations of sl(3). From (1.5.2) we find that (17.3.2) −s ∞ X ∞  X X m1 m2 (m1 + m2 ) −s ζW (s; sl(3)) = (dim φ) = . 2 φ m =1 m =1 1

2

This is because the irreducible representations of sl(3) are a family of representations Um1 ,m2 parametrized by m1 , m2 ∈ N, with 1 dim Um1 ,m2 = m1 m2 (m1 + m2 ). 2 Therefore from (17.3.2) we see that p2 (n) satisfies the generating function identity ∞ X n=0

n

p2 (n)X =

∞ Y m1 ,m2 =1

1 1−

X m1 m2 (m1 +m2 )/2

.

From this observation we may expect that ζW (s; sl(3)) will be useful to study p2 (n). In fact, inspired by the method developed in [131], Romik studied the analytic properties of ζ2 ((s, s, s); sl(3)) in detail, and (combining other ideas from asymptotic analysis, the theory of theta functions, probability theory etc.) he proved the following asymptotic formula: Theorem 17.7. (Romik [177]) As n → ∞, we have (17.3.3) p2 (n) =


K 2/5 3/10 1/5 1/10 exp A n − A n − A n − A n (1 + o(1)), 1 2 3 4 n3/5

17.4. A CONNECTION WITH SCHUR MULTIPLE ZETA-FUNCTIONS

395

where A1 = 5ξ 2 , A2 = ξ −1 η, A3 = (3/80)ξ −4 η 2 , A4 = (11/3200)ξ −7 η 3 and √   2 3π 1/3 1 −10 4 K = √ ξ exp − ξ η 2560 5 with  3/10 √ 1 2 ξ= Γ(1/3) ζ(5/3) , η = − πζ(1/2)ζ(3/2). 9 Further researches on the topic discussed in this section have been done recently by Bringmann et al. [22][23][24]. 17.4. A connection with Schur multiple zeta-functions Let λ = (λ1 , . . . , λm ) be a partition of n ∈ N, that is, {λi } is a non-increasing sequence of positive integers whose sum is equal to n. The Young diagram of λ is a collection of n square boxes, with λi boxes in the ith row (1 ≤ i ≤ m). This can be identified with the set D(λ) = {(i, j) ∈ Z2 | 1 ≤ i ≤ m, 1 ≤ j ≤ λi }.

Let X be a set, and make a Young tableau of the diagram by filling any box (i, j) with some element xij ∈ X. If X = N and the entries in each row are weakly increasing from left to right and those in each column are strictly increasing from top to bottom, we call it a semi-standard Young tableau of shape λ. Denote the set of all semi-standard Young tableaux of shape λ by SSTY(λ). The Schur multiple zeta-function of λ is defined by X 1 (17.4.1) ζλ (s) = , Ms M ∈SSYT(λ)

where Ms =

Y (i,j)∈D(λ)

s

mijij

for M = (mij ) ∈ SSYT(λ).

This notion was introduced by Nakasuji, Phuksuwan and Yamasaki [164]. When D(λ) is just one column, then ζλ (s) is nothing but the Euler– Zagier multiple zeta-function ζEZ,n (s11 , . . . , sn1 ). On the other hand, when D(λ) is just one row, then ζλ (s) becomes X 1 ⋆ (17.4.2) ζEZ,n (s11 , . . . , s1n ) = s11 s12 m1 m2 · · · msn1n 1≤m ≤···≤m 1

n

which is usually called the multiple zeta-star function. Special values of (17.4.2), called multiple zeta-star values (MZSV), are important objects

396

17. MISCELLANEOUS RESULTS

in the theory of multiple zeta values. Therefore Schur multiple zeta⋆ functions, which interpolate these two notions ζEZ,n and ζEZ,n , should also be interesting. A skew Young diagram is a diagram obtained as a set difference of two Young diagrams λ and µ, with µi ≤ λi for all i. In particular, when µ = (k, . . . , k) (ℓ-times) and λ = (k + 1, . . . , k + 1) ((ℓ + 1)-times), we call the corresponding diagram a Young diagram of anti-hook type, and denote it by rib(k|ℓ). The associated Young tableau can be illustrated as follows: mkℓ

.. , .

(17.4.3)

mk1 m00 m10 · · · mk0

where we renumbered the indices in boxes as above. The corresponding Schur multiple zeta-function ζrib(k|ℓ) (s) can be defined similarly as above. The second-named author and Nakasuji [142] discovered that ζrib(k|ℓ) (s) can be written in terms of a certain generalized form of zeta-functions of root systems of type A. For r > 0 and 0 ≤ d ≤ r, let (17.4.4) ∞ X

• ζr,d (s, Ar ) =

m1 =0

|

···

∞ X

!′

md =0

{z

d times

∞ X m

=1

···

} | d+1 {z

∞ X

Y

mr =1 1≤i 0, (17.4.5) ζrH (s, x, Ar ) =

∞ X m1 =1

···

∞ X

Y

mr =1 1≤i