Number Theory in Memory of Eduard Wirsing 3031316169, 9783031316166

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Number Theory in Memory of Eduard Wirsing Helmut Maier Jörn Steuding Rasa Steuding Editors

Number Theory in Memory of Eduard Wirsing

Helmut Maier • Jörn Steuding • Rasa Steuding Editors

Number Theory in Memory of Eduard Wirsing

Editors Helmut Maier Department of Mathematics University of Ulm Ulm, Germany

Jörn Steuding Institut für Mathematik University of Würzburg Würzburg, Germany

Rasa Steuding Institut für Mathematik University of Würzburg Würzburg, Germany

ISBN 978-3-031-31616-6 ISBN 978-3-031-31617-3 https://doi.org/10.1007/978-3-031-31617-3

(eBook)

Mathematics Subject Classification: 26D10, 41A10, 11N25, 11N37, P11P55, 11P32, 11K31, 11A05, 11L07, 11B50, 11N05, 11A41, 11A55, 11J70, 33C10, 11P32, 11M26, 11N37, 11A05, 11A25, 11N37, 11P21, 11M41, 34K43, 47H10, 47J25, 33E30, 11J72, 11J81, 11J86, 11A05, 11A25, 11P32, 11P55, 11N37, 11N25, 11N60, 11B75, 05A05, 05A16, 11A05, 11N45, 11P32, 11N99 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Dedicated to the memory of Eduard Wirsing

Preface

This book is intended to be a memorial to the mathematical work of our friend and colleague Eduard Wirsing. The dear reader will find in this volume some personal reminiscences of the exceptional mathematician who died on March 2022, an overview of the scientific work and many current research articles from the field of number theory. We begin with the survey article, followed by the personal recollections and finally the scientific contributions in alphabetical order. We are grateful to Nicola Oswald for the watercolor on the title page. For the provided photographs we thank Doris Schwarz and Władisław Narkiewicz. Furthermore, we would like to thank the anonymous reviewers as well as Gautami Bhowmik, Adriana Buic˘a, Rainer Dietmann, Titus Hilberdink, Athanasios Sourmelidis, Pascal Stumpf, Marc Technau. Finally, our thanks go to Springer Publishing House and Remi Lodh and Daniel Jagadisian in particular. Ulm, Germany Würzburg, Germany Würzburg, Germany December 2022

Helmut Maier Jörn Steuding Rasa Steuding

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Contents

Life and Work of Eduard Wirsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helmut Maier, Rasa Steuding, and Jörn Steuding

1

Remembering Eduard Wirsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lutz Gerhard Lucht

19

Personal Memories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helmut Maier

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On the Infimum of the Absolute Value of Successive Derivatives of a Real Function Defined on a Bounded Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . Michel Balazard

27

Friable Averages of Oscillating Arithmetic Functions . . . . . . . . . . . . . . . . . . . . . . . Régis de la Bretèche and Gérald Tenenbaum

43

Ein quaternäres Waring-Goldbach-Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jörg Brüdern

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Coprimality of Consecutive Elements in a Piatetski-Shapiro Sequence . . . Jean-Marc Deshouillers, Michael Drmota, and Clemens Müllner

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Wirsing’s Elementary Proofs of the Prime Number Theorem with Remainder Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Harold G. Diamond

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Diophantine Analysis Around .[1, 2, 3, . . . ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Carsten Elsner and Christopher Robin Havens On a Smoothed Average of the Number of Goldbach Representations . . . . 145 Daniel A. Goldston and Ade Irma Suriajaya Estimates for k-Dimensional Spherical Summations of Arithmetic Functions of the GCD and LCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Randell Heyman and László Tóth

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Contents

The Rational Points Close to a Space Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Martin N. Huxley Twists by Dirichlet Characters and Polynomial Euler Products of L-Functions, II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Jerzy Kaczorowski and Alberto Perelli Solving the Iterative Differential Equation .−γg  = g −1 . . . . . . . . . . . . . . . . . . . . . 223 Roland Miyamoto and Jürgen Sander Irrationality of Zeros of the Digamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 M. Ram Murty Generalizations of Menon’s Arithmetic Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Melvyn B. Nathanson On a Conjecture of Descartes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 János Pintz On the Greatest Common Divisor of a Number and Its Sum of Divisors, II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Paul Pollack Permutations with Arithmetic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Carl Pomerance Large Subsums of the Möbius Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Imre Z. Ruzsa The a-Points of the Riemann Zeta-Function and the Functional Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Athanasios Sourmelidis, Jörn Steuding, and Ade Irma Suriajaya Braided Gibonacci Sequences on Residue Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Jürgen Spilker and Luu Ba Thang

Life and Work of Eduard Wirsing Helmut Maier, Rasa Steuding, and Jörn Steuding

1 A Brief Biography On March 22, 2022, Eduard Wirsing passed away at the age of 90 in Cologne. He was an open-minded character who contributed great ideas to mathematics with his remarkable creativity. A man and his work that we will not forget. Eduard Wirsing was born in Berlin on June 28, 1931. His youth was overshadowed by the darkest side of German history. As a young adult Eduard was at the right age to be able to devote himself to his scientific interests in the newly rebuilding Germany after the terrible war. In 1950, he started his studies at the University of Göttingen and the Free University of Berlin and passed the teaching profession state examination in 1955. Two years later, Eduard Wirsing received his doctorate with a dissertation on essential components in additive number theory, supervized by Hans-Heinrich Ostmann (1913–1959) in Berlin. Then Dr. Wirsing worked as an assistant at the Technical University in Brunswick, not so far from Berlin, and finished his habilitation there in 1959. Since 1961 Wirsing was employed at the University of Marburg, where his doctoral supervisor Ostmann, who had just and suddenly passed away, had started his academic career. In the following years Eduard Wirsing would achieve his best and most influential results. This and the new place also proved to be a springboard to the wider world. In 1966/1967 Eduard Wirsing was visiting professor at the University of Nottingham, in 1967/1968, he worked at the renowned Cornell University, and in 1969 he was appointed professor in Marburg. In 1970/1971, he

H. Maier University of Ulm, Ulm, Germany e-mail: [email protected] R. Steuding · J. Steuding () Department of Mathematics, Würzburg University, Würzburg, Germany e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Maier et al. (eds.), Number Theory in Memory of Eduard Wirsing, https://doi.org/10.1007/978-3-031-31617-3_1

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was granted leave of absence for the Institute of Advanced Studies at Princeton. The next career jump was not long in coming, which was not surprising given the remarkable research results which we will discuss below in detail. Back in Germany, the path southwestward continued. In 1974 Wirsing accepted a call to the University of Ulm, where he remained until his retirement in 1999. Together with his colleagues Hans-Egon Richert (1924–1993) and Wolfgang Jurkat (1929– 2017) at Ulm, the medium-sized city on the Danube became a worldwide renowned center for number theory in the 1970s. The 15th Journées Arithmétiques with 148 participants was held in Ulm in 1987; the proceedings [49] were edited by Eduard Wirsing and Hans Peter Schlickewei. Wirsing was very involved in the promotion of young scientists. He served as a lecturer of the German National Academic Foundation,1 he organized numerous conferences at the Mathematical Research Institute Oberwolfach but also in Blaubeuren (near Ulm). He guided nine talents to doctorate,2 namely (chronologically listed) • • • • • • • • •

Dieter Gromes (1965) Reinhard Hermann (1965) Ilse Fuchs (1969) Hartmut Siebert (1970) Johann Goguel (1972) Reinhard Suck (1976) Johannes Schmid (1981) Luis Rocha (1994) Ulrike Vorhauer (1996)

Three of his pupils, Goguel, Siebert and Vorhauer, continued their academic careers. Behind every scientist there is also a human being. Eduard Wirsing was a person with many interests, and not only mathematical ones. He passionately played the alto flute and the board games chess and Go. He also had a penchant for tinkering and constructed imaginative and useful electronic devices. Creativity, passion and enthusiasm were always present in Eduard Wirsing’s actions. The second author recalls a conference in Palanga, Lithuania, in September 2006, where he presented a question on the Calkin-Wilf tree in the problem session. Wirsing was in the audience. A few days after the conference his solution to this recreational math problem was received, and it showed on the one hand his joy and enthusiasm even for simple problems. In what follows, however, we will focus on Eduard Wirsing’s seminal scientific work; the bibliography also includes articles that we do not address explicitly, or consider only in passing.

1 Vertrauensdozent

der Studienstiftung des Deutschen Volkes. to the Mathematics Genealogy Project, a service of the NDSU Department of Mathematics in association with the American Mathematical Society; see https://www.genealogy. math.ndsu.nodak.edu. 2 According

Life and Work of Eduard Wirsing

3

2 The Early Years 1950–1957: Additive Number Theory Additive number theory is concerned with the study of sets of integers and their behaviour under addition. Wirsing’s first papers dealt mostly with questions from this number-theoretical subdiscipline. In his very first publication, [69] from 1953, in the middle of his studies, Wirsing showed that almost all subsets of .N are totally primitive (or, in other words, asymptotically indecomposable). In order to explain the notion of total primitivity we quote from a review of Wirsing’s paper written by Paul Erd˝os (1913–1996) (Fig. 1) for the Mathematical Reviews here: Ostmann calls a sequence of integers .a1 < a2 < . . . primitive if there do not exist two sequences .b1 < b2 < . . . and .c1 < c2 < . . . each having more than one element so that the sequence .{bi + cj } consists precisely of the a’s (the same .ar may occur several times in the form .bi + cj ). The sequence is called totally primitive if every sequence which differs from it only in a finite number of terms is primitive.

This quote shows not only an early acknowledgement of Wirsing’s work by the about 20 years older Erd˝os but also explains the origin of the research question and the starting point of Wirsing’s interest in the work of his later thesis advisor Ostmann. The topic of Wirsing’s dissertation (entitled Über wesentliche Komponenten in der additiven Zahlentheorie) forms the content of his next publication [60]. If .A, B and W are subsets of .N ∪ {0}, and .A(x) and .B(x) count the positive elements of A

Fig. 1 Eduard Wirsing with Andrzej Schinzel (left) and Paul Erd˝os (middle) during a conference. Published with kind permission of Doris Schwarz

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and B not exceeding x, then W is said to be an essential component if the existence of the positive Schnirelmann density, A(n) ∈ (0, 1), n∈N n

σ (A) := inf

.

always implies that σ (A + W ) > σ (A).

.

Erd˝os [11] had shown that every basis is an essential component; recall that a subset B ⊂ N is called an additive basis if for some finite number k every positive integer can be written as a sum of k or fewer elements of .B . In joint work [60] with Alfred Stöhr (1916–1973), also from Berlin, an example of an essential component is given which is not an additive basis. The first such example was provided by Yuri Linnik (1915–1972) [35], however, the new example obtained was much simpler. In a later work, [86], Wirsing constructed for every . > 0 an essential component with counting function

.

  W (x)  exp (log x)1/2 log log log x ,

.

where .log stands for the natural logarithm here and in the sequel. Imre Ruzsa [45] improved this result by providing essential components with counting function 1+ . .W (x)  (log x) Of interest is also the multiplicative analogue. A sequence of strictly increasing positive integers .an is called a multiplicative basis of order k if every integer is a product of k or fewer .an ’s. In [71], Wirsing showed that in this case, for every .k ∈ N, .

lim inf x→∞

A(x) log x > 1, x

where .A(x) counts the numbers .an ≤ x. Moreover, he proved that, for every . > 0 there exists a multiplicative basis of order k satisfying .

lim inf x→∞

A(x) log x < 1 + . x

In 1934, Nikolai Romanov (1907–1972) [43] proved his celebrated theorem that given a fixed .b > 1, the set of numbers that can be represented as a sum of a prime and a positive integer power of b has a positive asymptotic density. This result was improved in a paper [20] by Bernhard Hornfeck (1929–2006), another pupil of Ostmann, slightly older than Wirsing. To formulate his result let .δ(A ) denote the asymptotic density of a set .A ⊂ N and denote by .f (A ) the set of all positive

Life and Work of Eduard Wirsing

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integers of the form .f (a) with .a ∈ A . Then Hornfeck showed that δ(P + f (N)) > 0

.

for every integer-valued polynomial f with positive leading coefficient, where .P denotes the set of prime numbers. A little later, Wirsing [72] succeeded in even showing that δ(P + f (P)) > 0

.

under the same conditions. The case of .f (x) = x implies the celebrated theorem of Lev Schnirelmann (1905–1938) [54] that the Minkowski sum .P + P has positive asymptotic density which also may be considered as a first approximation to the binary Goldbach conjecture. The case .f (x) = x 2 could have been obtained by Romanov’s method (as was mentioned by Erd˝os [12] without proof). The main new idea in Wirsing’s paper was, however, a uniform bound for the number of primes in an arithmetic progression found by Edward Charles Titchmarsh (1899–1963) (resulting from Brun’s sieve method) [62]. There are two joint papers [21, 22] with Hornfeck in this period. One of them deals with the old yet unsolved problem about the existence of odd perfect numbers, i.e., an odd integer n whose proper divisors add up to n; so far only even perfect numbers are known (such as 6 and 28). The two young mathematicians showed for the number .A(x) of odd perfect numbers .n ≤ x the bound    log x log log log x ; A(x) = O exp c log log x

.

in a later paper [74] Wirsing removed the triple logarithm in the numerator. Problems of an additive nature were always on the mind of Eduard Wirsing. In a later paper [90], Wirsing proved that one can choose from every additive basis of order k of .N satisfying some technical conditions a subbasis of order k with counting function of order .O((x log x)1/k ). Since the squares form a basis of order .k = 4, as follows from the classic four square theorem of Joseph-Louis Lagrange (1736–1813), this result of Wirsing improved a previous bound by J. Zöllner [98].

3 The Postdoc Period 1958–1969: Primes and Approximations The time after the doctorate is possibly the most important for a researcher. This period was probably exceptionally formative and groundbreaking for our protagonist (Fig. 2).

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Fig. 2 The young Eduard Wirsing as he jumps from one discipline to the next, making valuable contributions each time. Published with kind permission of Władisław Narkiewicz

Prime numbers are the multiplicative particles of the integers. The distribution of prime numbers has fascinated generations of mathematicians. The first quantitative result in this direction is the divergence of the sum of the reciprocals of the primes due to Leonhard Euler (1707–1783) which he noted as .

1 1 1 1 + + + + . . . = log log ∞ . 2 3 5 7

Edmund Landau [32] used this formula as the starting point for a study on how many integers have all their prime factors in a given arithmetic progression. Generalizing this result, Wirsing [70] proved that for a set .T of primes an asymptotic formula of the form .

 1 = τ log log x + c1 + o(1) p

T p≤x

with constants .τ, c1 implies that the counting function of positive integers composed of primes in .T is asymptotically equal to .c2 x(log x)τ −1 with another constant .c2 . Landau [32] was dealing with the case when .T is the union of arithmetic progressions. A hot topic in the early years after World War II were the elementary proofs of the prime number theorem found by Erd˝os and (not independently) Atle Selberg (1917–2007); see [14, 57]. If .π(x) denotes, as usual, the counting function for all

Life and Work of Eduard Wirsing

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primes .p ≤ x, then the existence of the following limit .

lim π(x) ·

x→∞

log x =1 x

is called the prime number theorem; that the limit in the case of its existence is equal to 1 follows already by partial summation from classical results such as Euler’s asymptotical formula above. The earlier proofs of the prime number theorem all rely on analytic properties of the Riemann zeta-function and methods from complex analysis or advanced techniques from real analysis (so-called Tauberian theorems). A good read on the remarkable work of Erd˝os and Selberg and their controversy is Melvyn Nathanson’s textbook [40]. The discovery of an elementary proof— not making any use of any deeper concepts of analysis than the logarithm and inequalities—was a big surprise. By refinement of the methods of Erd˝os and Selberg, first P. Kuhn [29] and Johannes van der Corput (1890–1975) [64], then Enrico Bombieri [3] and, a little later, Eduard Wirsing [77, 78] gave elementary proofs of the prime number theorem with an error term of order .O(x/(log x)B ), where .B > 0 is arbitrary. In the meantime the error terms achieved by elementary means have been improved; the best so far is due to W.C. Lu [36] who obtained an error .O(x exp(−c(log x)α )) for every .α < 12 .3 Arithmetical functions play a central role in number theory. Of special interest are multiplicative arithmetical functions. It was conjectured by Erd˝os and Aurel Wintner (1903–1958) that every multiplicative function f assuming only the values .0, ±1 possesses a mean-value, that is the existence of the limit M(f ) := lim

.

x→∞

1 f (n). x n≤x

One of the first examples that comes into mind in this context is the Möbius .μfunction which (generates the coefficients of the Dirichlet series representation of the reciprocal of the zeta-function and) has mean-value zero. That this vanishing mean-value is equivalent to the prime number theorem indicates the depth of this question. It was Eduard Wirsing who proved this conjecture of Erd˝os and Wintner with his celebrated mean-value theorem: if f is a multiplicative real-valued function f satisfying .|f (n)| ≤ 1, then its mean-value always exists and is equal to    1 f (p) f (p2 ) 1− 1+ + + ... , .M(f ) = lim x→∞ p p p2 p≤x

3 In this volume the dear reader can find with Harold Diamond’s contribution a detailed account of Wirsing’s work.

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where the product is considered to be zero if the series .

 1 − f (p) p

p

diverges (as, for example, in the case .f = μ). The related papers [76, 80] of Wirsing thus provide a new proof of the prime number theorem (and therefore may be regarded as a continuation of his earlier works [77, 78]). However, Wirsing’s theorem, as is often the case, did not end investigations on the subject, but spurred further research. The case of complex-valued functions f of modulus .|f (n)| ≤ 1 was already considered by Wirsing [80]; however, the final solution was found by Gábor Halász [17] only one year later; he proved the existence of a real constant .α, a complex constant c, and a slowly oscillating function .L : [1, ∞) → C such that  .

f (n) = c · L(log x) · x 1+iα + o(x);

n≤x

the constants as well as the function L may be given explicitly (in terms of f ), and slowly oscillating here means that, for every .A > 0, the limit .limx→∞ L(Ax)/L(x) exists and is equal to 1. Both mean-value theorems, Wirsing’s as well as Halász’s result, appear in numerous research monographs, for example in Wolfgang Schwarz (1934–2013) and Jürgen Spilker’s monograph [55] or Peter Elliott’s two volumes on probabilistic number theory [10]. The related papers [76, 80] by Wirsing are his most cited overall. An important work on Diophantine approximation also falls into this phase after the doctorate. In 1955, Karl-Friedrich Roth (1925–2015) made a significant progress in determining the approximation quality of algebraic numbers by rational numbers and improving earlier results of Axel Thue (1863–1922), Carl Ludwig Siegel (1896– 1981) and Freeman Dyson (1923–2020) [44]. For this proof of a conjecture of Siegel (and for his works on progression-free sets and the theory of irregularities of distribution) Roth was awarded the Fields Medal in Edinburgh 1958. Reason enough for young ambitious scientists to follow in these footsteps. Two years later Wirsing [75] showed that given a real algebraic number .ξ and a positive integer d, for any . > 0 there exist only finitely many algebraic numbers .α of degree d such that |ξ − α| < H (α)−2d− ,

.

where .H (α) denotes the height of .α (that is the maximum of the absolute values of the minimum polynomial of .α over .Z). The case .d = 1 is Roth’s celebrated theorem. Interestingly, the general case follows from this with the help of certain inequalities originating from probability theory. It follows that algebraic numbers cannot be approximated too well by algebraic numbers of any degree. This beautiful

Life and Work of Eduard Wirsing

9

piece of work is one of the most cited works by Wirsing. In the meantime, the exponent .−2d− could be improved to .−d−1− by Wolfgang Schmidt [52]. A rich theory of algebraic numbers has developed, while our knowledge of transcendental numbers is rather fragmentary. About a decade later Wirsing [82] took up this direction of research again and showed, among other things, that the classifications of the transcendental numbers of Kurt Mahler (1903–1988) and Jurjen Koksma (1904–1964) from the 1930s actually coincide. For these classifications and Wirsing’s work we refer to [1, 53]. Wirsing’s work with two other extremely well-known mathematicians also occurred during this period. Alan Baker (1939–2018) had been awarded a Fields Medal in Nice in 1970 for his explicit estimates of linear forms in logarithms and Brian John Birch made outstanding contributions to the theory of elliptic curves, not the least of which was the conjecture he made jointly with Sir Peter Swinnerton–Dyer (1927–2018), which is now one of the six remaining unsolved millennium problems. In joint work with Baker and Birch [2], a conjecture of Savardaman Chowla (1907–1995) from the context of special values of L-functions was addressed; this problem dates back to the late 1940s and some exchange with Siegel but is first present in the literature only in 1970 with Chowla’s work [7]. Let f be a q-periodic arithmetical function with values in the field .Q of algebraic numbers. Moreover, suppose that .f (n) = 0 whenever .1 < gcd(n, q) < q,  f (n) .

n≥1

n

= 0,

and the cyclotomic polynomial . q is irreducible over .Q(f (1), . . . , f (q)). Then, under these assumptions, it was shown that f vanishes identically. A crucial ingredient was Baker’s bound for linear forms in logarithms. Also, explicit solutions were given when one of the conditions is deleted. It was conjectured by Erd˝os that under the assumptions .f (q) = 0 and .f (n) = ±1 for .n ≡ 0 mod q the series is non-vanishing (cf. [39]). For this, more recent work on this open conjecture, and how this is related to the work of Baker, Birch and Wirsing, we refer to a more recent paper [39] by Ram Murty and Siddhi Pathak.4 A joint work [28] with Hans Günther Kopetzky and Wolfgang Schwarz a few years later also falls into this context. Generalizing a result of M. Newman [41], the authors proved that a set of inequalities  .

j ρjk ≥ 0

for k = 1, 2, 3, . . . ,

1≤j ≤n

where .ρ1 , . . . , ρn are distinct elements of the circle group T = {z ∈ C : |z| = 1}

.

4 See

also Ram Murty’s contribution in this volume.

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and arbitrary .1 , . . . , n ∈ T , implies that the .ρj ’s are the n-th roots of unity and ρj → j is a group homomorphism. The reasoning relies on a theorem of Leopold Kronecker (1823–1891) on the denseness of the vectors .(kβ1 , . . . , kβn ) modulo 1 for .k = 1, 2, . . ., where .β1 , . . . , βn are linearly independent over the rationals. Also the case of distinct characters .χ1 , . . . , χn of a compact abelian group was considered. A further contribution to this area was a joint work [50] with Hans Peter Schlickewei (from Marburg, the former domain of Eduard Wirsing). They provided estimates for the heights going beyond previous bounds of S. Zhang [97] and Don Zagier [96] and applied their results to the linear Diophantine equation

.

aX + bY = 1.

.

This work was improved later by Enrico Bombieri and Umberto Zannier [4], Wolfgang Schmidt [51], and others. In a joint paper [5] with Jerzy Browkin (1934– 2015) Wirsing returned to some part of this topic.

4 Thirty Years as Professor 1969–1999: From Logarithms to Partitions The time of Wirsing’s appointment as professor also saw the beginning of another topic that was to occupy him throughout the rest of his life. Motivated by an old paper of Erd˝os [13] Wirsing was considering characterizations of the omnipresent logarithm as an additive function. In [81], he proved that if f is additive and satisfies f (n + 1) = f (n) + O(1)

.

for all positive integers n, then .f (n) = c log n + O(1) with some constant c. And in [83] Wirsing proved a conjecture of Erd˝os, namely, that if f is additive and if  .

|f (n + 1) − f (n)| = o(x),

n≤x

then .f (n) = c log n with some constant c. An independent proof of this result was given by Imre Kátai [25]. Further results of this type were established in [61, 87–89]. Multiplicative counterparts were studied in [63, 92, 95], some in joint work with others. It may be said that the theory of arithmetic functions developed simultaneously with Wirsing’s academic career into an independent and beautiful sub-discipline of number theory with many important results from his pen. Wirsing’s interests were widely spread as the next examples show. Continued fractions have been extremely useful tools for Diophantine approximation since ancient times. Their measure-theoretic properties were already studied by Carl Friedrich Gauss (1777–1855) in a diary entry with date October 25, 1800, which

Life and Work of Eduard Wirsing

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is about a century before the development of measure theory itself. Let .measn (x) denote the Lebesgue measure of those real numbers .α ∈ (0, 1) with continued fraction expansion .α = [a0 , a1 , a2 , . . . , an , an+1 , . . .] satisfying [an , an+1 , . . .] − an < x.

.

Then the theorem of Gauss-Kuzmin-Lévy states that, for every .x ∈ (0, 1), .

lim measn (x) =

n→∞

log(1 + x) . log 2

This already allows to deduce results about the statistics of the partial quotients an (e.g., that the geometric mean of the .an ’s converges almost surely to a limit .≈ 2.68). In a letter to Pierre-Simon Laplace (1749–1827) in 1812, Gauss asked for a quantitative version, which later became known as the Gauss problem. The first published proof of the existence of the limit is due to Rodion Kuzmin (1891–1949) in 1928 who also provided an answer to Gauss’ problem [31]. One year later Paul Lévy (1886–1971) [34] came up with an independent proof and an improved error estimate. In his famous Art of Computer Programming, Donald Knuth wrote .

“Gauss’s problem was really resolved until 1974, when Eduard Wirsing published a beautiful analysis of the situation”

(see [26], page 363). Indeed, the best possible error estimate was determined by Wirsing [84, 85] in showing that measn (x) −

.

 log(1 + x) = (−x)n (x) + O x(1 − x)μn , log 2

where .μ, λ are constants satisfying .0 < μ < λ = 0.30366 . . . and . is a smooth positive function. Wirsing’s proof relies on a functional equation for the measure and a generalization of a classical result of Georg Frobenius (1849–1917) on the spectrum of positive linear operators [15]; “the simplest aspects of Wirsing’s approach” are very well explained in [26], pp. 363. Nowadays, these results form the foundations of the ergodic theory of continued fractions. Partitions have been studied for centuries. In particular the work of Srinivasa Ramanujan (1887–1920) has significantly influenced many number theorists and has also led to investigations of partitions from a wide variety of perspectives. In [48], a joint work of Eduard Wirsing with Andrzej Schinzel (1937–2021), partitions are studied in context of the arithmetical function .ω(n) counting the number of distinct prime factors of n. If .p(m) denotes the number of partitions of m, then it is shown that ⎛ ⎞  1− log N .ω ⎝ p(m)⎠ > log 2 m≤N

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for every sufficiently large N (depending on . > 0). This implies that for infinitely many integers n the largest prime factor of .p(n) is larger than .log n. In this context Javier Cilleruelo (1961–2016) and Florian Luca [8] showed that for almost all n, the largest prime factor of .p(n) is larger than .log log n.

5 Retirement and No End: Lattice Points, Rigidity, and More Retirement means parting with some duties and opens up new freedoms. In the years around Wirsing’s retirement, there is no sign of a declining research activity visible. The young Gauss counted the points .(a, b) with integer coordinates inside a circle by comparing them with the area of the circle. This simple but ingenious geometric argument implies  .

1

r2 (n) − π x  x 2 ,

0≤n≤x

where .r2 (n) counts the number of integer solutions to the equation .a 2 + b2 = n and the error term is of the size of the radius of the circle. The search for the smallest possible error term is known as the circle problem. The present best error term estimate is .O(x 131/416 (log x)1.13 ) due to Martin Huxley [23]. Concerning the question how small such an error can be, a classical result of Godfrey Hardy (1877– 1947) and Edmund Landau (1877–1938) showed independently that the exponent in the error term estimate cannot be smaller than . 14 [19, 33]. In the works [66, 67], Wirsing in joint work with Ulrike Vorhauer presented a new proof of a celebrated result [6] of Chen Jingrun (1933–1996) for the logarithmic Riesz mean, namely, .

  κ  12 25  1 r2 (n) log(x/n) = π x + O x 37 − 37 κ+ (k + 1) n≤x

7 and every positive ., where the gamma factor on the for every non-negative .κ ≤ 30 left is a normalization with respect to the Riesz mean (or the log-powers in the sum). The approach of Vorhauer and Wirsing was using two-dimensional Weyl steps, an advanced tool from the theory of exponent pairs. The second part [65] of the series of papers is a single-authored work of Vorhauer dealing with the logarithmic Riesz mean for a class of arithmetical functions. A further joint paper with Vorhauer considered a conjecture of Peter Sarnak in the context of the Selberg class. Recall that in 1989 Selberg [58] introduced an axiomatic setting for all zeta- and L-functions appearing in number theory. Roughly

Life and Work of Eduard Wirsing

13

speaking, the Selberg class .S consists of all meromorphic functions .L that (1) possess a convergent Dirichlet series representation L (s) =



.

a(n)n−s

n≥1

in some right half-plane, (2) have coefficients satisfying .a(n)  n , (3) have an analytic continuation to the whole complex plane except for at most a pole at m .s = 1 such that .(s − 1) L (s) is an entire function of finite order with some appropriate positive integer m, (4) satisfy a Riemann-type functional equation, and, finally, (5) possess an Euler product (with certain restrictions on the coefficients in order to prevent counterexamples to an analogue of the Riemann Hypothesis for .L ). Obvious elements in .S are the Riemann zeta-function, Dirichlet L-functions .L(s; χ ) associated with a primitive character .χ , their shifts .L(s + iτ ; χ ) for any real number .τ , and Dedekind zeta-functions. Of special interest is the structure of the Selberg class .S (which is a multiplicative semi-group). An element .L ∈ S is said to be primitive if a factorization .L = L 1 L 2 within .S implies that one of the factors is identical 1. A weak version of Selberg’s (in general unproven) orthonormality conjecture claims for two primitive elements .L 1 , L 2 ∈ S that  a1 (p)a2 (p)  (1 + o(1)) log log x if L 1 = L 2 , = . o(log log x) otherwise, p p≤x where the .aj (p) are the Dirichlet series coefficients of .L j (s) on the primes p. Another open conjecture in this context states that there are only countably many shift classes of primitive functions in the Selberg class, where two elements .L 1 , L 2 ∈ S are said to be shifted, if .L 2 (s) = L 1 (s + iτ ) for some real number .τ . This notion is motivated by the observation that with an entire function .L (s) also every shift .L (s + iτ ) with a real number .τ is an element of .S . A continuous family of functions in .S on an interval .I ⊂ R is a set of functions .L (s; ξ ), ξ ∈ I , where .L(s; ξ ) ∈ S for every .ξ and .L(s; ξ ) is continuous with respect to .ξ ; in a similar manner one defines a continuous family of primitive functions in .S with the requirement that .L (s; ξ ) is always primitive. Then, Sarnak’s rigidity conjecture states that (1) any continuous primitive family on an interval .I ⊂ R is of the form .L (s; ξ ) = L (s + ih(ξ )) with a primitive .L and a continuous function .h : I → C, and (2) any continuous family can be factored into primitive continuous families. In [68], Vorhauer and Wirsing showed that the orthonormality conjecture and the countability conjecture imply Sarnak’s rigidity conjecture. Interestingly, their reasoning relies on an old topological result [59] of Wacław Sierpi´nski (1882–1969). An earlier and different proof of this result had been given by Jerzy Kaczorowski and Alberto Perelli [24].

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Another research topic of Wirsing after his retirement were cyclotomic polynomials; two of the three corresponding articles [27, 30, 93] are joint work with the next generation of number theorists.

6 Isolated, Unpublished and Late Results The curse of time: in an early paper [73], Wirsing showed that every convex curve with two equichordal points has a regular boundary. In 1997, Marek Rychlik [46] proved that there is no such curve. In the interesting research monograph [38] on the interface between analytic number theory and harmonic analysis one can find a subsection on Gorškov– Wirsing polynomials in the final chapter on small polynomials with integral coefficients. The author, Hugh Montgomery, remarks that his “exposition follows the unpublished work of E. Wirsing, who was unaware of the earlier work of Gorškov” [16]. Wirsing’s contribution from 1981 resp. 2003 includes an irreducibility criterion for a certain sequence of polynomials defined by a quadratic recurrence. The last published article of Eduard Wirsing appeared in a commemorative volume for the late Wolfgang Schwarz in 2016. In this short note [94] Wirsing gave a simple and short proof of a theorem of Hardy [18] on the existence of power series which converge uniformly on the unit circle but not absolutely. This result answered a question posed by Marcel Riesz (1886–1969). It should be noted that there were earlier papers on power series by Wirsing, namely [37, 79]. During his more than sixty years lasting research period Eduard Wirsing published altogether 45 publications. Some do not fit into any of our boxes, e.g., [42, 56, 63, 91] although each of them contains very original ideas of interesting mathematics. As Freeman Dyson said Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking and bring together diverse problems from different parts of the landscape. Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time.

(see [9]). In this classification, Eduard Wirsing was, in our opinion, more a frog than a bird, maybe a flying frog whose solutions to big problems made significant contributions to the development of postwar mathematics up to the present day. Acknowledgments The authors are indebted to the late Andrzej Schinzel and Wolfgang M. Schmidt for their excellent presentation of Eduard Wirsing’s scientific work on the occasion of his 75th birthday [47]; this work was very helpful for this survey. The authors are also grateful to Doris Schwarz and Władisław Narkiewicz for providing the photographs.

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References 1. A. Baker, Transcendental Number Theory (Cambridge University Press, 1975) 2. A. Baker, B.J. Birch, E. Wirsing, On a problem of Chowla. J. Number Theory 5, 224–236 (1973) 3. E. Bombieri, Maggiorazione del resto nel “Primzahlsatz” col metodo di Erd˝os-Selberg. Ist. Lombardo Accad. Sci. Lett. Rend. A 96, 343–350 (1962) 4. E. Bombieri, U. Zannier, Algebraic points on subvarieties of Gnm . Int. Math. Res. Not. 7, 333– 347 (1995) 5. J. Browkin, E. Wirsing, Rank two matrices with elements of norm 1. Funct. Approx. 33, 7–14 (2005) 6. J.-R. Chen, The lattice points in a circle. Sci. Sin. 12, 633–649 (1963) 7. S. Chowla, The nonexistence of nontrivial linear relations between the roots of a certain irreducible equation. J. Number Theory 2, 120–123 (1970) 8. J. Cilleruelo, F. Luca, On the largest prime factor of the partition function of n. Acta Arith. 156, 29–38 (2012) 9. F. Dyson, Birds and Frogs. Not. Am. Math. Soc. 56, 212–223 (2009) 10. P.D.T.A. Elliott, Probabilistic Number Theory, vol. I (Springer, 1979); vol. II (Springer, 1980) 11. P. Erd˝os, On the arithmetical density of the sum of two sequences one of which forms a basis for the integers. Acta Arith. 1, 197–200 (1936) 12. P. Erd˝os, On the asymptotic density of the sum of two sequences one of which forms a basis for the integers, II. Trav. Inst. Math. Tbilissi 3, 217–224 (1938) 13. P. Erd˝os, On the distribution function of additive functions. Ann. Math. 47, 1–20 (1946) 14. P. Erd˝os, On a new method in elementary number theory which leads to an elementary proof of the prime number theorem. Proc. Nat. Acad. Sci. U S A 35, 374–384 (1949) 15. G. Frobenius, Über Matrizen aus positiven Elementen, Sitzungsber. königl. Preuss. (Akademie d. Wissensch., Berlin, 1908), pp. 471–476 16. D.S. Gorškov, On the distance from zero on the interval [0, 1] of polynomials with integral coefficients, in Proceedings of the Third All Union Mathematical Congress, Moscow 1956, vol. 4, (Akad. Nauk SSSR, Moscow, 1959), pp. 5–7 17. G. Halász, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen. Acta Math. Acad. Sci. Hung. 19, 365–403 (1968) 18. G.H. Hardy, A theorem concerning Taylor’s series. Q. J. Pure Appl. Math. 44, 147–160 (1913) 19. G.H. Hardy, On the expression of a number as the sum of two squares. Q. J. Math. 46, 263–283 (1915) 20. B. Hornfeck, Zur Struktur gewisser Primzahlsätze. J. Reine Angew. Math. 196, 156–169 (1956) 21. B. Hornfeck, E. Wirsing, Über die schwache Basisordnung. Arch. Math. 7, 450–452 (1956) 22. B. Hornfeck, E. Wirsing, Über die Häufigkeit vollkommener Zahlen. Math. Ann. 133, 431–438 (1957) 23. M.N. Huxley, Exponential sums and lattice points III. Proc. London Math. Soc. 87, 591–609 (2003) 24. J. Kaczorowski, A. Perelli, On the structure of the Selberg class, III, Sarnak’s rigidity conjecture. Duke Math. J. 101, 529–554 (2000) 25. I. Kátai, On a problem of P. Erd˝os. J. Number Theory 2, 1–6 (1970) 26. D.E. Knuth, The Art of Computer Programming, vol. 2, 3rd edn. (Addison Wesley, 1997) 27. S. Konyagin, H. Maier, E. Wirsing, Cyclotomic polynomials with many primes dividing their orders. Period. Math. Hungar. 49, 99–106 (2004) 28. H.G. Kopetzky, W. Schwarz, E. Wirsing, Nonnegative sums of roots of unity. Arch. Math. 34, 114–121 (1980) 29. P. Kuhn, Eine Verbesserung des Restgliedes beim elementaren Beweis des Primzahlsatzes. Math. Scand. 3, 75–89 (1955) 30. M. Künzer, On coefficient valuations of Eisenstein polynomials. J. Théorie Nombres Bordeaux 17, 801–823 (2005)

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31. R.O. Kuzmin, On a problem of Gauss. Dokl. Akad. Nauk SSSR, 375–380 [Russian] (1928) 32. E. Landau, Lösung des Lehmer’schen Problems. Amer. J. Math. 31, 86–102 (1909) 33. E. Landau, Über die Gitterpunkte in einem Kreise (Nachr. Ges. Wiss., Göttingen, 1915), pp. 148–160; 209–243 34. P. Lévy, Sur les lois de probabilité dont dépendent les quotients complets et incomplets d’une fraction continue. Bull. Soc. Math. France 57, 178–194 (1929) 35. Yu.V. Linnik, On Erd˝os’ theorem on the addition of numerical sequences. Math. Sb. 10, 67–78 (1942) 36. W.C. Lu, On the elementary proof of the prime number theorem with a remainder term. Rocky Mt. J. Math. 29, 979–1053 (1999)  37. W. Miesner, E. Wirsing, On the zeros of (n + 1)k zn . J. Lond. Math. Soc. 40, 421–424 (1965) 38. H.L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis (American Mathematical Society, 1994) 39. M.R. Murty, S. Pathak, The Okada space and vanishing of L(1, f ). Funct. Approx. 66, 35–57 (2022) 40. M. Nathanson, Elementary Methods in Number Theory (Springer, 2000) 41. M. Newman, Nonnegative sums of roots of unity. J. Res. Nat. Bur. Stand. B 80, 1–4 (1976) 42. A. Peyerimhoff, E. Stickel, E. Wirsing, On the rate of convergence for two term recursions. Computing 40, 329–335 (1988) 43. N. Romanov, Über zwei Sätze der additiven Zahlentheorie. Math. Ann. 109, 668–678 (1934) 44. K.F. Roth, Rational approximations to algebraic numbers. Mathematika 2, 1–20 (1955); corrigendum p. 168. 45. I. Ruzsa, Essential components. Proc. Lond. Math. Soc. 54, 38–56 (1987) 46. M.R. Rychlik, A complete solution to the equichordal point problem of Fujiwara, Blaschke, Rothe and Weizenböck. Invent. Math. 129, 141–212 (1997) 47. A. Schinzel, W.M. Schmidt, The mathematical work of Eduard Wirsing. Funct. Approx. 35, 7–18 (2006) 48. A. Schinzel, E. Wirsing, Multiplicative properties of the partition function. Proc. Indian Acad. Sci. Math. Sci. 97, 297–303 (1987) 49. H.P. Schlickewei, E. Wirsing (eds.), Number Theory, Proceedings of the 15th journées arithmétiques Held in Ulm, 1987, Lecture Notes in Mathematics, vol. 1380 (Springer, 1989) 50. H.P. Schlickewei, E. Wirsing, Lower bounds for the heights of solutions of linear equations. Invent. Math. 129, 1–10 (1997) 51. W.M. Schmidt, Linear forms with algebraic coefficients, I. J. Number Theory 3, 253–277 (1971) 52. W.M. Schmidt, Heights of points on subvarieties of Gnm . Lond. Math. Soc. Lect. Notes Ser. 235, 157–187 (1996) 53. T. Schneider, Einführung in die transzendenten Zahlen (Springer, 1957) 54. L. Schnirelmann, Über additive Eigenschaften von Zahlen. Ann. Inst. polytechn. Novoˇcerkask 14, 3–28 (1930); Math. Ann. 107, 649–690 (1933) 55. W. Schwarz, J. Spilker, Arithmetical Functions (Cambridge University Press, 1994) 56. W. Schwarz, E. Wirsing, The maximal number of non-isomorphic abelian groups of order n. Arch. Math. 24, 59–62 (1973) 57. A. Selberg, An elementary proof of the prime number theorem. Ann. Math. 50, 305–313 (1949) 58. A. Selberg, Old and new conjectures and results about a class of Dirichlet series, in Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), vol. 2 (1992), pp. 47–63 59. W. Sierpi´nski, Un theorème sur les continues. Tôhoku Math. J. 11, 300–303 (1918) 60. A. Stöhr, E. Wirsing, Beispiele von wesentlichen Komponenten, die keine Basen sind. J. Reine Angew. Math. 196, 96–98 (1956) 61. Y.-S. Tang, P.-T. Schao, E. Wirsing. On a conjecture of Kátai for additive functions. J. Number Theory 56, 391–395 (1996) 62. E.C. Titchmarsh, A divisor problem. Rend. Circ. Mat. Palermo 54, 414–429 (1930); corrected in 57, 478–479 (1933)

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63. L. Toth, E. Wirsing, The maximal order of a class of multiplicative arithmetical functions. Ann. Univ. Sci. Budapest Sect. Comput. 22, 353–364 (2003) 64. J.G. Van der Corput, Sur le reste dans la démonstration élémentaire du théorème des nombres premiers, in Colloque sur la Théorie des Nombres, Bruxelles, 1955 (Georges Thone, Liège; Masson & Cie, Paris, 1956) 65. U.M. Vorhauer, Three two-dimensional Weyl steps in the circle problem, II: the logarithmic Riesz mean for a class of arithmetic functions. Acta Arith. 91, 57–73 (1999) 66. U.M. Vorhauer, E. Wirsing, Three two-dimensional Weyl steps in the circle problem, I: the Hessian determinant. Acta Arith. 91, 43–55 (1999) 67. U.M. Vorhauer, E. Wirsing, Three two-dimensional Weyl steps in the circle problem, III: exponential Integrals and applications, in Number Theory in Progress, vol. 2 (1999), pp. 1131– 1146. 68. U.M. Vorhauer, E. Wirsing, On Sarnak’s rigidity conjecture. J. Reine Angew. Math. 531, 35–47 (2001) 69. E. Wirsing, Ein metrischer Satz über Mengen ganzer Zahlen. Arch. Math. 4, 392–398 (1953) 70. E. Wirsing, Über die Zahlen, deren Primteiler einer gegebenen Menge angehören. Arch. Math. 7, 263–272 (1956) 71. E. Wirsing, Über die Dichten multiplikativer Basen. Arch. Math. 8, 11–15 (1957) 72. E. Wirsing, Eine Erweiterung des ersten Romanovschen Satzes. Arch. Math. 9, 407–409 (1958) 73. E. Wirsing, Zur Analyzität von Doppelspeichenkurven. Arch. Math. 9, 300–307 (1958) 74. E. Wirsing, Bemerkung zu der Arbeit über vollkommene Zahlen. Math. Ann. 137, 316–318 (1959) 75. E. Wirsing, Approximation mit algebraischen Zahlen beschränkten Grades. J. Reine Angew. Math. 206, 67–77 (1960) 76. E. Wirsing, Das asymptotische Verhalten von Summen über multiplikative Funktionen. Math. Ann. 143, 7–103 (1961) 77. E. Wirsing, Elementare Beweise des Primzahlsatzes mit Restglied. I. J. Reine Angew. Math. 211, 205–214 (1962) 78. E. Wirsing, Elementare Beweise des Primzahlsatzes mit Restglied. II. J. Reine Angew. Math. 214/215, 1–18 (1964) 79. E. Wirsing, On the monotonicity of the zeros of two power series. Michigan Math. J. 13, 215– 218 (1966) 80. E. Wirsing, Das asymptotische Verhalten von Summen über multiplikative Funktionen II. Acta Math. Acad. Sci. Hungar. 18, 411–447 (1967) 81. E. Wirsing, A characterization of log n as an additive arithmetic function, in Symposia Math., vol. IV (INDAM, Rome, 1968/1969) (Academic Press, London, 1970), pp. 45–47 82. E. Wirsing, On approximation of algebraic numbers by algebraic numbers of bounded degree, in 1969 Number Theory, Proc. Symp. Pure Math., vol. XX (State Univ. New York/Stony Brook/American Mathematical Society, 1971), pp. 213–247 83. E. Wirsing, Characterization of the logarithm as an additive function, in 1969 Number Theory, Proc. Symp. Pure Math., vol. XX (State Univ. New York/Stony Brook/American Mathematical Society, 1971), pp. 375–381 84. E. Wirsing, Über den Satz von Gauß-Kusmin-Lévy, Ber. Math. Forschungsinstitut Oberwolfach, No. 5, (Bibliographisches Institut, Mannheim, 1971), pp. 229–231 85. E. Wirsing, On the theorem of Gauss-Kusmin-Lévy and Frobenius type theorem for function spaces. Acta Arith. 24, 507–562 (1973/1974) 86. E. Wirsing, Thin essential components, in Topics in Number Theory, Colloq. Math. Soc. Janos Bolyai, vol. 13 (1976), pp. 429–492 87. E. Wirsing, Additive functions with restricted growth on the numbers of the form p + 1. Acta Arith. 37, 345–357 (1980) 88. E. Wirsing, Additive and completely additive functions with restricted growth, in Recent Progress in Analytic Number Theory, vol. 2 (Academic Press, 1981), pp. 231–280 89. E. Wirsing, Growth and differences of additive arithmetic functions, in Topics in Number Theory, vol. 2. Colloq. Math. Soc. Janos Bolyai, vol. 34 (1984), pp. 1651–1661

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90. E. Wirsing, Thin subbases. Analysis 6, 285–308 (1986) 91. E. Wirsing, Functions without residue and a bilinear differential equation. Acta Arith. 64, 157– 174 (1993) 92. E. Wirsing, On a problem of Kátai and Subbarao. Ann. Univ. Sci. Budapest Sect. Comput. 24, 69–78 (2004) 93. E. Wirsing, The third logarithmic momentum of the cyclotomic polynomial on the unit circle and factorizations with a linear side condition, in Proc. ElAZ Conference, May 24–28, 2004 (Franz Steiner Verlag, Stuttgart, 2006), pp. 297–312 94. E. Wirsing, A minimal proof of a result of Hardy, in From Arithmetic to Zeta-Functions, ed. by J.W. Sander et al. (Springer, 2016), pp. 523–526 95. E. Wirsing, D. Zagier, Multiplicative functions with difference tending to zero. Acta Arith. 100, 75–78 (2001) 96. D. Zagier, Algebraic numbers close to both zero and 1. Math. Comput. 61, 485–491 (1993) 97. S. Zhang, Positive line bundles on arithmetic surfaces. Ann. Math. 136, 569–587 (1992) 98. J. Zöllner, Über eine Vermutung von Choi, Erd˝os and Nathanson. Acta Arith. 45, 211–213 (1985)

Remembering Eduard Wirsing Lutz Gerhard Lucht

Eduard Wirsing will remain in my memory as a great role model, both as a mathematician and as a person. His quick comprehension, enormous breadth of knowledge, speed of reaction, spontaneity and friendly attention have impressed me again and again. Long reflected insights he liked to present when he thought it was worth convincing his counterpart. In doing so, he was never concerned with argument, but rather with conviction, whereas by inquiries and contradictions, he stayed calm and benevolent. Until his death, he was and remained a philanthropist who had internalised the miracle of life and believed in the good. I had met Eduard Wirsing at the end of the 1960s when he lectured at the Technical University (TU) Braunschweig at the invitation of his Berlin student friend Bernhard Hornfeck at the TU Braunschweig and at the after-session in Mr Hornfeck’s flat, where he discussed with me the Erd˝os’ mean value problem, the solution of which also provided a novel proof of the prime number theorem. At the same time, he encouraged me to read up, after my forthcoming doctorate, by considering two works still in print by Gábor Halász and Wolfgang Schwarz, onto the relevant field of research on multiplicative functions. I did indeed follow his recommendation and have written several papers on the subject, among them my habilitation thesis at the Technical University (TU) Clausthal, for which he served as an external reviewer. In the time that followed, I met him frequently. We met regularly at the international Oberwolfach conferences on elementary and analytic number theory organized by him, Hans-Egon Richert, and Wolfgang Schwarz. After the restructuring of the Oberwolfach conferences in 2003, I was able—with the support of my then assistant Christian Elsholtz (now in Graz) and doctoral and diploma students—to start a new series of conferences for the promotion of young scientists, with changing venues. The number of international participants has now grown far beyond that of the first and only

L. G. Lucht () Goslar, Germany © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Maier et al. (eds.), Number Theory in Memory of Eduard Wirsing, https://doi.org/10.1007/978-3-031-31617-3_2

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sparsely funded conference entitled Elementare und Analytische Zahlentheorie (ELAZ) at TU Clausthal. In 1966/1967, Wirsing had a visiting professorship at the University of Nottingham and lectured on the mean value problem for complex-valued multiplicative functions which was shortly afterwards surprisingly solved by the Turán student Gabor Halász from Budapest; at my request the Nottingham lecture notes were sent to me (free of charge). A little later, I met Eduard Wirsing again by chance at an annual meeting of the German Mathematical Association (DMV). He seemed extremely cheerful and agile, and during the lunch break he suggested that instead of going to the restaurant reserved for the meeting, we should go better to a little restaurant further away and invited me to join him. The food was really very good, he ordered a bottle of wine and I asked if there was a special occasion, like a birthday. He smiled and said no, referring only to the “the good conversations” he had earlier, and refused when I tried to pay for my meal, saying that I could pay him back when I had finished my apprenticeship and was earning well. In addition to Wirsing’s important contribution on the mean values of real-valued multiplicative functions, I had also become interested early on in his work on perfect numbers. It was proposed by Erd˝os to show that the set of perfect numbers has zero density. After the first proof of this open problem by Hornfeck and Wirsing [1], Wirsing [2] had proved a quantitative statement in an elementary and very elegant way by showing that the number of solutions .n ≤ x of .σ (n) = κ · n for (rational) .κ > 0 is smaller than   log x (x ≥ 3), . exp c · (1) log log x where C is a certain positive independent of .κ and .σ denotes the divisor sum function. In 1971, in a dissertation at the TU Braunschweig, among other things, a corresponding estimate for the solutions of .f (n) = 2 · n was proposed, where .f = σ ◦ σ is the concatenation of the divisor sum function .σ with itself. For the proof it was stated that only Wirsing’s proof needed to be transferred. However, since this concatenation f is no longer multiplicative, Wirsing’s argument alone cannot work. Since the doctoral student himself could not close the gap and had also left academia, I finally wrote a paper1 in which the error was resolved and a more general statement was proven: Theorem Let k be a natural number .≥ 1 and .f = fk ◦· · ·◦f1 be the concatenation of multiplicative functions .f : N → N, which have certain general asymptotic properties. Then the number of solutions of the equation .f (n) = κ · n is also below the above Wirsing’s bound (1).

1 Lutz Lucht, Über die Hintereinanderschaltung multiplikativer Funktionen, J. Reine Angew. Math., Vol. 283/284 (1976), 275–281.

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The idea of proof is to transform the equation .f (n) = κ · n into a system of equations .f (n ) = n+1 for . = 0, 1, ..., k−1 with .n0 = n and to consider .nk = κ · n. Then every solution n of the equation .f (n) = κ · n is also the solution of the product equation  .

1≤≤k

f (n ) =



n .

1≤≤k

This is a purely multiplicative equation for k-dimensional vectors .(n1 , ..., nk ) ∈ Nk . The proof therefore amounts to a k-dimensional application of Wirsing’s prime factorization of the product .n1 · · · nk . It suffices to treat the case .k = 2, what remains simply follows by iteration, which amounts to products of Wirsing’s bounds. For this, the conditions on the class of functions .f must ensure that all quotients . fp(p) lie in a sufficiently small neighbourhood of 1, the growth of the functions .f at the  higher prime powers is restricted, and, finally, the product equation . 1≤≤k f (a ) =  a does not produce, apart from the trivial solution, solutions whose all  1≤≤k prime divisors exceed a positive bound (and therefore cannot appear as solution of the equation .f (n) = κ · n). Furthermore, it turns out that Wirsing’s bound is even best possible for the class of functions formed from the concatenations of multiplicative functions as it was shown in a subsequent joint paper.2 Examples of such concatenations for .k = 2 include, besides .σ ◦ σ , for instance .σ ◦ ϕ, .ϕ ◦ σ or .ϕ ◦ ϕ, where .ϕ is the Euler .ϕ-function. I very fondly remember the conference on the occasion of the 70th birthday of Eduard Wirsing in Ulm, at which I was allowed to give a keynote lecture on inversion and Ramanujan expansion of arithmetic functions; he noticed very attentively and gratefully the new methods and results that were presented. All the guests, including myself, were generously entertained by him. It should be noted that Mr. Wirsing himself was extremely thrifty and modest, having witnessed the times of need in divided Berlin, when his academic teacher Hans-Heinrich Ostmann tried to help his starving and freezing doctoral students and staff, including Eduard Wirsing and Bernhard Hornfeck, with “surplus food”. As a native Berliner, I had experienced this misery myself as a child. One of Ostmann’s particularly successful ideas consisted of reviewing pathological attempts to solve unsolved or insoluble problems. These included the angle division with ruler and compass, squaring the circle, the calculation of the currently largest prime number or prime number twins as well as other geometric, combinatorial, logical or set-theoretical problems. As I had learned from Bernhard Hornfeck, Ostmann held well-attended introductory lectures in private rooms, which he was able to obtain rent-free from interested parties, in which he and his colleagues presented numerous problems and noted that the proposed solutions would be

Lucht, Wolfgang Schwarz, Über die Lösungsanzahl der Geichung .f (n) = κ · n für gewisse Klassen multiplikativer Funktionen, Monatshefte Math., Vol. 81 (1976), 213–216.

2 Lutz

22

L. G. Lucht

carefully examined scientifically. Unfortunately, this was so time-consuming that a moderate fee had to be paid for it. The fee was to be paid in kind, such as potatoes, butter, lard, eggs, meat, cabbage, turnips, apples, wood, coal, candles, light bulbs, paper, pens, etc. The campaign was extremely successful and there were enough customers. The correction was quite precise and the problem solvers received encouraging advice such as: “That’s better than last time, keep it up!” or “You need to work harder on [this or that part] to improve your accuracy!” In fact, I would have had the opportunity much earlier, namely in the summer semester of 1962 at the TU Braunschweig at the beginning of my mathematics studies, to participate on a course entitled “Introduction to Mathematics” held by Wirsing. At that time he and Mr. Hornfeck were doctoral assistants at the Institute A of Ordinarius X (anonymised), which was located in a barrack near Spielmannstraße. However, since I was studying mathematics and physics, I had to choose the course with the same name given by some experimental physicist. Both courses took place in parallel. Only once I was able to attend Wirsing’s course unnoticed. When I went to the blackboard to ask Mr Wirsing a question about the lecture, the control tutor caught me and noted me down as an unauthorised intruder. “Lecture fee fraud” was punishable by law. First I had to repent by going to Ordinary X, from whom I and a number of other fellow students received the snide remark, without any hearing. “If you can’t read, then you’d better learn a trade or become an architect!” This was followed by a complaint of fraud to the chancellor of the TU Braunschweig, from whom I received a request for a written statement, then a “blue letter” with a summons to be questioned by the chancellor on the grounds of hearing fee fraud and possible de-registration. After waiting for an hour with trembling knees and a chalky face, the chancellor’s secretary told me to go home, that the matter was closed, the chancellor had better things to do and I would get off scot-free, the case was dropped. I am sure that Mr Wirsing would have commented appropriately on the events of the time in this anecdote. With the onset of education reforms in the 1960s and 1970s, which also affected universities, student fees were converted into general tuition fees. The collection of tuition fees at mass events had to be prohibited in a series of individual decrees. This led to a major wave of lawsuits from established full professors with conflicting rulings until a general academic reform put an end to it. At the TU Brauschweig, which had been badly damaged in the World War, there were greater expansions in terms of buildings and personnel, which had a correspondingly positive effect on teaching and research in the field of mathematics. Decades later I learned, that Ordinarius X assigned in 1936 at the TH Braunschweig, since the first university reforms in the 1970s sued by all authorities against the abolition of tuition fees with a moderate increase in salaries, and ultimately lost. For me, that was a satisfaction and I saw it as a victory for justice: it is legitimate to ask professional questions without being criminalised for it. And I had met Mr Wirsing after all. Eduard Wirsing had a long, fulfilled life and was mentally alert and physically active until his death. His son Peter told me that he persevered in resisting all his illnesses, even with artificial joints and an implant that largely eliminated his hearing loss. On the day of his death, when he returned home from early morning and

Remembering Eduard Wirsing

23

shopping, he had sat down to breakfast in the morning as usual when, out of the blue, he was struck by a stroke. Deeply rooted in his faith, he was torn out of his earthly existence. He had never feared it and knew that it could happen at any time. Besides his son, he leaves behind two daughters and his life mate of his later years, to whom my sincere condolence goes. I will always remember him fondly and am sad that with him and the already deceased Wolfgang Schwarz (Frankfurt), who died in 2013, my both significant scientific role models, colleagues and fatherly friends are no longer alive.

Published with kind permission of Władisław Narkiewicz

Published with kind permission of Władisław Narkiewicz

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L. G. Lucht

References 1. B. Hornfeck, E. Wirsing: über die Häufigkeit der vollkommenen Zahlen. Math. Ann. 133, 431– 438 (1957) 2. E. Wirsing, Bemerkung zu der Arbeit über vollkommene Zahlen in Math. Ann. Bd. 133, 431– 438 (1957). Math. Ann. 137, 316–318 (1959)

Personal Memories Helmut Maier

The first time I was meeting Professor Eduard Wirsing was in Summer 1974. Back then I was studying for the diploma in mathematics with physics as minor subject. His lecture on Analytic Number Theory was in late afternoon after the “Praktikum” belonging to the physics course. After the experiment we had to perform, being a not very practical person, his lecture came to me as a liberation. After my graduation from the University of Ulm and a short era as a scientific employee I was performing my doctoral studies and part of my later career in the United States. During that time I had no contact to Professor Wirsing. In 1993, when I became Professor In Ulm, I got to know him as a very kind and helpful colleague who helped me to adjust to the new environment. Our scientific collaboration was not particularly wide but together with Sergei Konyagin we studied cyclotomic polynomials, an important object of Number Theory. We established a result, saying that cyclotomic polynomials with many primes dividing their orders always have large coefficients. I also had opportunity to experience another side of Professor Wirsing: his remarkable general education. Once, during a conference in Colfosco in Alto Adige, we had a walk together in the Dolomites. At that time I had taken an interest in the development of chemistry and by some chance I brought up the topic of Stahl’s theory of phlogistone (negative oxygen) from the early eighteenth century. Professor Wirsing had a profound knowledge of this topic. For the travel linked to the conference we were sharing a car. When we were passing near Innsbruck, Professor Wirsing talked about Andreas Hofer’s uprising against Napoleon. Whenever somebody tried to start a conversation about a certain subject there was a good chance that Professor Wirsing had something to say about it.

H. Maier () University of Ulm, Ulm, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Maier et al. (eds.), Number Theory in Memory of Eduard Wirsing, https://doi.org/10.1007/978-3-031-31617-3_3

25

26

H. Maier

There was of course also a downside to this. Professor Wirsing was very interested in philosophy and as a consequence knew a lot about philosophers whereas my knowledge about this subject was basically non-existent. Once we unsuccessfully tried to start a talk on it. If someone had suggested we should rather talk about the weather, we probably both gladly would have followed this advice. Professor Wirsing had also been a longtime member of the Doctoral Committee. Many times when he heard about a thesis defence—the themes could be of the most diverse kinds—he enthusiastically rushed to attend the colloquium. The death of Professor Wirsing is a painful loss—especially also for the scientific community.

Published with kind permission of Doris Schwarz

On the Infimum of the Absolute Value of Successive Derivatives of a Real Function Defined on a Bounded Interval Michel Balazard

To the memory of Eduard Wirsing, master of analysis, and of its applications to number theory.

Abstract A study of the greatest possible ratio of the smallest absolute value of a higher derivative of some function, defined on a bounded interval, to the .Lp -norm of the function. Keywords Chebyshev polynomials · Legendre polynomials · Extremal problems · Inequalities for derivatives

1 Introduction Let n be a positive integer, .I = [a, b] a bounded segment of the real line, of length L = b−a. Define .D n (I ) as the set of real functions f defined on I , with successive derivatives .f (k) defined and continuous on I for .0 ≤ k ≤ n − 1, and .f (n) defined on .I˚ = ]a, b[. We will use the notation

.

mn (f ) = inf |f (n) (t)|.

.

a 0 implies that .f (n) has constant sign on I , so that D ∗ (n, p, λ, I ) = inf{f p , f ∈ D n (I ), f (n) (t) ≥ λ for a < t < b}.

.

Thus, determining .C ∗ (n, p, I ) is equivalent to minimizing .f p for .f ∈ D n (I ) with the constraint .f (n) (t) ≥ λ > 0 for .a < t < b. We will denote this extremal problem by .E ∗ (n, p, λ, I ).

3 The Relevance of Monic Polynomials Let .P n be the set of monic polynomials of degree n, with real coefficients, identified with the set of the corresponding polynomial functions on I , which is a subset of n .D (I ). Since .mn (f ) = n! for .f ∈ P n , one has D ∗ (n, p, n!, I ) ≤ D ∗∗ (n, p, I ),

.

(2)

where D ∗∗ (n, p, I ) = inf{Qp , Q ∈ P n }.

.

A basic fact in the study of the extremal problem .E ∗ (n, p, λ, I ) is that (2) is in fact an equality. Proposition 3.1 For all .n, p, I , one has .D ∗ (n, p, n!, I ) = D ∗∗ (n, p, I ). It follows from this proposition that .C ∗ (n, p, I ) = n!/D ∗∗ (n, p, I ) and, by (1), C(n, p) = Ln+1/p n!/D ∗∗ (n, p, I ).

.

(3)

Let us review the history of Proposition 3.1. For .p = ∞, it is a corollary to a theorem of S. N. Bernstein from 1937. Denoting by .Ek (f ) the distance (for the uniform norm on I ) between f and the set of

30

M. Balazard

polynomials of degree at most k, he proved in particular that En−1 (f0 ) > En−1 (f1 )

.

(f0 , f1 ∈ D n (I )),

(n) (n) provided that the inequality .f0 (ξ ) > |f1 (ξ )| is valid for every .ξ ∈ I˚ (cf. [2], p. 48, inequalities (47bis)–(48bis)). Proposition 3.1 follows by taking .f1 (x) = x n and n (n) (t) ≥ n! for .f0 (x) = λf (x), where f is a generic element of .D (I ) such that .f .a < t < b, and .λ > 1, then letting .λ → 1. This theorem of Bernstein was generalized by Tsenov in 1951 to the case of the p .L -norm on I , where .p ≥ 1 (cf. [15], Theorem 4, p. 477), thus providing a proof of Proposition 3.1 for .p ≥ 1. The case .0 < p < 1 was left open by Tsenov. The study of the extremal problem .E ∗ (n, p, λ, I ) was one of the themes of the 1993 PhD thesis of Xiaoming Huang [5]. In Lemma 2.0.7, pp. 9–10, she gave another proof (due to Saff) of Proposition 3.1 in the case .p = ∞. For .1 ≤ p < ∞, she gave a proof of Proposition 1 which is unfortunately incomplete (cf. [5], pp. 28–30). Again, the case .0 < p < 1 was left open. We present now a self-contained proof of Proposition 3.1, valid for .0 < p ≤ ∞. As it proceeds by induction on n, we will need the following classical-looking division lemma, for which we could not locate a reference (compare with [16] or [13]).

Proposition 3.2 Let .n ≥ 2 and .f ∈ D n (I ). Let .c ∈ [a, b]. Put  g(x) =

.

f (x)−f (c) x−c f (c)

(x ∈ I, x = c) . (x = c)

(4)

Then .g ∈ D n−1 (I ). For every .x ∈ ]a, b[ , one has g (n−1) (x) =

.

f (n) (ξ ) , n

where .ξ ∈ ]a, b[. Proof Since .f is continuous, one has 

1

g(x) =

.

  f c + t (x − c) dt

(x ∈ I ).

0

Using the rule of differentiation under the integration sign, one sees that g is .n−2 times differentiable on I , with  g

.

(n−2)

1

(x) =

  t n−2 f (n−1) c + t (x − c) dt

(x ∈ I ).

0

As .f (n−1) is continuous on I , this formula yields the continuity of .g (n−2) on I .

On the Infimum of the Absolute Value of Successive Derivatives

31

The function g is n times differentiable on .I˚ \ {c} (this set is just .I˚ if .c = a or .c = b), being a quotient of n times differentiable functions, with non-vanishing denominator. In the case .a < c < b, we have now to check that g is .n − 1 times differentiable at the point c. The function .f (n−1) being continuous on I and differentiable at the point c, there exists a function .ε(h), defined and continuous on the segment .[a − c, b − c] (the interior of which contains 0), vanishing for .h = 0, such that f (n−1) (c + h) = f (n−1) (c) + hf (n) (c) + hε(h) (a ≤ c + h ≤ b).

.

Hence, 

1

g (n−2) (x) =

.



  t n−2 f (n−1) c + t (x − c) dt

0 1

=

   t n−2 f (n−1) (c) + t (x − c)f (n) (c) + t (x − c)ε t (x − c) dt

0

f (n−1) (c) f (n) (c) + (x − c) + (x − c) = n−1 n



1

  t n−1 ε t (x − c) dt

0

When x tends to c, the last integral tends to 0, so that the function .g (n−2) is differentiable at the point c, with g (n−1) (c) =

.

f (n) (c) · n

If .x ∈ I˚ \ {c}, one may use the general Leibniz rule and Taylor’s theorem with the Lagrange form of the remainder in order to compute .g (n−1) (x): g (n−1) (x) =

.

 1  d n−1  f (x) − f (c) · n−1 x−c dx

  (−1)n−1 (n − 1)! = f (x) − f (c) · (x − c)n

n−1  n − 1 (k) (−1)n−1−k (n − 1 − k)! f (x) · + k (x − c)n−k k=1

  f (k) (x) (n − 1)!  (c − x)k f (c) − f (x) − n (c − x) k! n−1

=

k=1

32

M. Balazard

=

(n − 1)! f (n) (ξ ) (c − x)n · (c − x)n n!

=

f (n) (ξ ) , n

where .ξ belongs to the open interval bounded by c and x.



In the next proposition, we stress the main element of our proof of Proposition 3.1, namely the fact that the condition .f (n) ≥ n!, for some .f ∈ D n (I ), implies that the absolute value of f dominates the absolute value of some monic polynomial of degree n. Proposition 3.3 Let .n ≥ 1 and .f ∈ D n (I ) such that .f (n) (x) ≥ n! for every .x ∈ ]a, b[ . Then there exists a monic polynomial P of degree n, with all its zeros in I , such that the inequality .|f (x)| ≥ |P (x)| is valid for every .x ∈ I . Moreover, if .|f (x)| = |Q(x)| for every .x ∈ I , where Q is a monic polynomial of degree n with real coefficients, then .f (x) = Q(x) for every .x ∈ I . Proof The assertion about the zeros may be obtained a posteriori, by replacing the zeros of P by their projections on I . The following proof leads directly to a polynomial P with all zeros in I . We use induction on n. For .n = 1, the function f is continuous on .[a, b], differentiable on .]a, b[ , with

.f (x) ≥ 1 for .a < x < b. If .f (a) ≥ 0, one has, for .a < x ≤ b, .f (x) = f (a) + (x − a)f (ξ ) (where .a < ξ < x), thus .f (x) ≥ x − a. Hence, one has .|f (x)| ≥ |x − a| for every .x ∈ I . If .f (b) ≤ 0, one proves similarly that .|f (x)| ≥ |x − b| for every .x ∈ I . If .f (a) < 0 < f (b), there exists .c ∈ ]a, b[ such that .f (c) = 0. One has then, for every .x ∈ I , f (x) = f (x) − f (c) = (x − c)f (ξ ) (where a < ξ < b).

.

Hence .|f (x)| ≥ |x − c| for every .x ∈ I , and the result is proven for .n = 1. Let now .n ≥ 2, and suppose that the result is valid with .n − 1 instead of n. Let .f ∈ D n (I ) such that .f (n) (x) ≥ n! for every .x ∈ ]a, b[ . If f vanishes at some point .c ∈ I , it follows from Proposition 2 that the function g defined on I by  g(x) =

.

f (x) x−c f (c)

(x ∈ I, x = c) (x = c)

On the Infimum of the Absolute Value of Successive Derivatives

33

belongs to .D n−1 (I ) and that, for every .x ∈ ]a, b[ , one has g (n−1) (x) =

.

f (n) (ξ ) , n

where .ξ ∈ ]a, b[, thus .g (n−1) (x) ≥ (n − 1)!. By the induction hypothesis, there exists a monic polynomial Q of degree .n − 1, with all its roots in I , such that .|g(x)| ≥ |Q(x)| for every .x ∈ I . Hence, one has the inequality .|f (x)| ≥ |P (x)| for every .x ∈ I , where .P (x) = (x − c)Q(x) is a monic polynomial of degree n, with all its roots in I . If .f > 0, it reaches a minimum at some point .c ∈ I . Again, it follows from Proposition 2 that the function g defined on I by (4) satisfies the required hypothesis for degree .n−1. Thus there exists a monic polynomial Q of degree .n−1, with all its roots in I , such that .|g(x)| ≥ |Q(x)| for every .x ∈ I . Hence, one has the inequality f (x) − f (c) = |f (x) − f (c)| ≥ |P (x)|

.

(x ∈ I ),

where .P (x) = (x − c)Q(x). It follows that .

|f (x)| = f (x) ≥ f (c) + |P (x)| > |P (x)|

(x ∈ I )

If .f < 0, the reasoning is similar by considering a point .c ∈ I where f reaches a maximum. Let us prove the last assertion. The hypothesis .|f | = |P | is equivalent to the equality .f 2 = P 2 , that is .(f −P )(f +P ) = 0. The set .E = {x ∈ I, f (x)+P (x) = 0} has empty interior, since .f (n) (x) + P (n) (x) = 0 on every open subinterval of E, whereas .f (n) (x) + P (n) (x) ≥ 2n! on .I˚. The set .I \ E is therefore dense in I ; its elements x all verify .f (x) = P (x), hence .f = P on I by continuity.  Proposition 3.1 is an immediate corollary of Proposition 3.3: by taking f and P as stated there, one has .|f (x)| ≥ |P (x)| for every .x ∈ I , so that  .

a

b

 |f (x)|p dx ≥

b

|P (x)|p dx,

(5)

a

for every .p > 0 (for .p = ∞: .max |f | ≥ max |P |). Moreover, if .p < ∞, equality in (5) implies that .|f | = |P | on I , hence .f = P . In other words, if .0 < p < ∞, the extremal problem .E ∗ (n, p, n!, I ) has exactly the same solutions (value of the infimum and extremal functions) as the problem ∗∗ .E (n, p, I ) obtained by considering only monic polynomials of degree n, which one may even take with all their roots in I . For .p = ∞, our reasoning does not prove that an extremal function for ∗ .E (n, p, n!, I ) (if it exists) must be a polynomial. This is true anyway, as proved by Huang in [5], pp. 10–13.

34

M. Balazard

4 Extremal Polynomials One may now use the results of the well developed theory of the extremal problem E ∗∗ (n, p, I ) for polynomials. Thus, since the integral

.

 .

b

|(x − x1 ) · · · (x − xn )|p dx

(x1 , . . . , xn ∈ I )

a

(or the value .maxx∈I |(x − x1 ) · · · (x − xn )|) is a continuous function of (x1 , . . . , xn ), the compactness of .I n yields the existence of an extremal (polynomial) function for .E ∗∗ (n, p, I ), hence for .E ∗ (n, p, n!, I ). It is a known fact that the polynomial extremal problem .E ∗∗ (n, p, I ) has a unique solution for all .p ∈ ]0, ∞], but there is no proof valid uniformly for all values of p.

.

• For .p = ∞, uniqueness was proved by Young in 1907 (cf. [18], Theorem 5, p. 340) and follows from the general theory of uniform approximation (cf. [12], Theorem 1.8, p. 28). • For .1 < p < ∞, as proved by Jackson in 1921 (cf. [6], §6, pp. 121–122), this is a consequence of the strict convexity of the space .Lp (I ). • For .p = 1, this is also due to Jackson in 1921 (cf. [7], §4, pp. 323–326). • For .0 < p < 1, the uniqueness of the extremal polynomial was proved in 1988 by Kroó and Saff (cf. [10], Theorem 2, p. 184). Their proof uses the uniqueness property for .p = 1 and the implicit function theorem. We will denote by .Tn,p,I the unique solution of the extremal problem E ∗∗ (n, p, I ). Uniqueness gives immediately the relation

.

Tn,p,I (a + b − x) = (−1)n Tn,p,I (x)

.

(x ∈ R).

Another property of these polynomials is the fact that all their roots are simple. For .p = 1, this fact was proved by Korkine and Zolotareff in 1873 (cf. [8], pp. 339– 340), before their explicit determination of the extremal polynomial (see Sect. 5.4 below), and their proof extends, mutatis mutandis, to the case .1 < p < ∞. For .p = ∞, this is a property of the Chebyshev polynomials of the first kind (see Sect. 5.2 below). Lastly, for .0 < p < 1, this was proved by Kroó and Saff in [10], p. 187. Define .Tn,p = Tn,p,[−1,1] , and write .n = 2k + ε, where .k ∈ N and .ε ∈ {0, 1}. It follows from the mentioned results that Tn,p (x) = x ε (x 2 − xn,1 (p)2 ) · · · (x 2 − xn,k (p)2 )

.

where 0 < xn,1 (p) < · · · < xn,k (p) ≤ 1.

.

(x ∈ R),

(6)

On the Infimum of the Absolute Value of Successive Derivatives

35

Kroó, Peherstorfer and Saff have conjectured that all the .xn,k are increasing functions of p (cf. [9], p. 656, and [10], p. 192).

5 Results on C(n, p) 5.1 The Case n = 1 The value .n = 1 is the only one for which .C(n, p) is explicitly known for all p. Proposition 5.1 One has .C(1, p) = 2(p + 1)1/p for .0 < p < ∞, and .C(1, ∞) = 2. Proof By (6), one has .T1,p (x) = x, so that, for .0 < p < ∞, D ∗∗ (1, p, [−1, 1]) =



1

.

−1

|t|p dt

1/p

 1/p = 2/(p + 1) ,

and, by (3), C(1, p) = 21+1/p /D ∗∗ (1, p, [−1, 1]) = 2(p + 1)1/p .

.

 Note that the Lemma 1.1, p. 6 of [11], asserts that .C(1, p) ≤ 2 · 31/p for .p ≥ 2, and that bound is .< 2(p + 1)1/p for .p > 2.

5.2 The Case p = ∞ This is the classical case, solved by Chebyshev in 1853 by introducing the polynomials .Tn defined by the relation .Tn (cos t) = cos nt (now called Chebyshev polynomial of the first kind): the unique solution of the extremal problem .E ∗∗ (n, ∞, [−1, 1]) is .21−n Tn . Let us record a short proof of this fact. Take .I = [−1, 1] and suppose that P is  a monic polynomial of degree n satisfying the inequality .P ∞ ≤ 21−n Tn ∞ = 21−n . Then, for .λ > 1 the polynomial Qλ = λ21−n Tn − P

.

is of degree n, with leading coefficient .λ − 1. Moreover, it satisfies (−1)k Qλ (cos kπ/n) = λ21−n − (−1)k P (cos kπ/n) > 0 (k = 0, . . . , n)

.

36

M. Balazard

By the intermediate value property, .Qλ has at least n distinct roots, hence exactly n, and these roots, say .x1 , . . . , xn , have absolute value not larger than 1. Hence, .

|Qλ (x)| = (λ − 1) |(x − x1 ) · · · (x − xn )| ≤ (λ − 1)(1 + |x|)n

(x ∈ R).

When .λ → 1, .Qλ (x) tends to 0 for every real x, which means that .P = 21−n Tn . One deduces from this theorem the value of .C(n, ∞). One has 1−n ∗∗ .D (n, ∞, [−1, 1]) = max 2 Tn (x) = 21−n , |x|≤1

hence C(n, ∞) = 2n · n!/D ∗∗ (n, ∞, [−1, 1]) = 22n−1 n!

.

(7)

(compare with the upper bound .C(n, ∞) ≤ 2n(n+1)/2 nn of [4], 3 (a), p. 185). This result is essentially due to Bernstein (cf. [1], p. 65). Qualitatively, the result expressed by (7) was nicely described by Soula in [14], p. 86, as follows. Bernstein’s principle: the minimum of the absolute value of the n-th derivative of an n times differentiable function and the maximum of the absolute value of the n-th derivative of an analytic function have similar orders of magnitude.

5.3 The Case p = 2 In this case, the extremal problem .E ∗∗ (n, 2, [−1, 1]) is an instance of the general problem of computing the orthogonal projection of an element of a Hilbert space onto a finite dimensional subspace. Here, the Hilbert space is .L2 (−1, 1), the element is the monomial function .x n , and the subspace is the set of polynomial functions of degree less than n. The solution follows from the theory of orthogonal polynomials: the extremal polynomial for .E ∗∗ (n, 2, [−1, 1]) is .

2n (n!)2 Pn (x) (2n)!

(|x| ≤ 1),

where .Pn is the n-th Legendre polynomial, defined by Pn (x) =

.

1 dn 2 (x − 1)n . 2n n! dx n

Hence, 2n (n!)2 2n (n!)2 Pn 2 = .D (n, 2, [−1, 1]) = (2n)! (2n)! ∗∗

2 , 2n + 1

On the Infimum of the Absolute Value of Successive Derivatives

37

(see [17], §15.·14, p. 305) and 1

C(n, 2) = 2n+ 2 · n!/D ∗∗ (n, 2, [−1, 1]) =

.

(2n)! √ 2n + 1, n!

(8)

a result given by Soula in 1932 (cf. [14], pp. 87–88).

5.4 The Case p = 1 The problem .E ∗∗ (n, 1, [−1, 1]) was solved by Korkine and Zolotareff in [8]: the extremal polynomial is .2−n Un (x), where .Un is the n-th Chebyshev polynomial of the second kind, defined by the relation .Un (cos t) = sin(n + 1)t/ sin t. Therefore, one has D ∗∗ (n, 1, [−1, 1]) = 2−n



1

.

= 2−n

−1  π

|Un (x)| dx = 2−n



π 0

|sin(n + 1)t| dt = 2−n

0

|Un (cos t)| sin t dt 

π

sin u du 0

= 21−n , and C(n, 1) = 2n+1 · n!/D ∗∗ (n, 1, [−1, 1]) = 22n n!.

.

(9)

5.5 Bounds for C(n, p) We begin with a simple monotony result. Proposition 5.2 For every positive integer n, the function .p → C(n, p) is decreasing on the interval .0 < p ≤ ∞. Proof Let .I = [0, 1]. Equivalently, we will see that the function .p → D ∗∗ (n, p, I ) is increasing. This is due to the fact that, for a fixed .f ∈ L∞ (I ) such that .|f | is not equal almost everywhere to a constant, the function .p → f p is increasing (a consequence of Hölder’s inequality). Thus, for every .Q ∈ P n and

.0 < p < p ≤ ∞, .

Qp > Qp ≥ D ∗∗ (n, p, I ),

which implies that .D ∗∗ (n, p , I ) > D ∗∗ (n, p, I ).



38

M. Balazard

In particular, (7) and (9) yield the inequalities 22n−1 n! < C(n, p) < 22n n! (1 < p < ∞).

.

The next proposition implies that the limit of .C(n, p) when p tends to 0 is (2e)n n!.

.

Proposition 5.3 For every positive integer n and every positive real number p, one has 2n (1 + np)1/p n! ≤ C(n, p) ≤ (2e)n n!

.

Proof Equivalently, we will prove that (2e)−n ≤ D ∗∗ (n, p, I ) ≤ 2−n (1 + np)−1/p ,

.

(10)

where .I = [0, 1]. Let .Q(t) = (t − x1 ) · · · (t − xn ), where .0 ≤ x1 , . . . , xn ≤ 1. One has  1 1 |Q(t)|p dt . ln Qp = ln p 0   1 1  ≥ ln |Q(t)|p dt by Jensen’s inequality p 0  1 = ln |Q(t)| dt 0

=

n  1 

ln |t − xk | dt.

k=1 0

Now, 

1

.

ln |t − x| dt = (1 − x) ln(1 − x) + x ln x − 1 (0 ≤ x ≤ 1),

0

attains its minimal value, namely .−1 − ln 2, when .x = 1/2. This implies the first inequality of (10). p To prove the second inequality of (10), we just compute .Qp when .Q(t) = n (t − 1/2) :  .

0

1

|t − 1/2|np dt = 2

(1/2)np+1 · np + 1 

On the Infimum of the Absolute Value of Successive Derivatives

39

For .0 < p < 1, we can also prove the following result. Proposition 5.4 Let n be a positive integer, and p such that .0 < p < 1. One has 1≤

.

1 C(n, p) ≤ (8/π )1/p . 2 22n n!

Proof The first inequality is just .C(n, 1) ≤ C(n, p). To prove the second inequality, let r and s such that .1 < s < 2 and .r −1 +s −1 = 1. Define  I1 (s) =

1

dt (1 − t 2 )s/2

.

−1

 I2 (s) =

1

|t|(s−1)/s √

−1

dt 1 − t2

·

The integrals .I1 (s) and .I2 (s) may be computed, using the Eulerian identity 

1

.

t x−1 (1 − t)y−1 dt = B(x, y) =

0

(x)(y) (x + y)

(x > 0, y > 0).

The results are (1 − 2s )2 (2 − s)  1  1 ( 2 )  1 − 2s

I1 (s) = 21−s

.

I2 (s) =

( 32 −

1 2s )

·

Now, let .Q ∈ P n and put .p = p/r. By Hölder’s inequality, one has  .

 dt

|Q(t)|p √ ≤ 1 − t2 −1 1



1

−1



|Q(t)|p r dt

1/r  

1

−1

1/s dt (1 − t 2 )s/2

p

= Qp I1 (s)1/s . It was proved by Kroó and Saff (cf. [10], pp. 182–183) that (n−1)p

2

.



1

−1

|Q(t)|

p

√

dt 1 − t2



1

dt

|Tn (t)|p √ 1 − t2 −1  π 

|cos nu|p du = = ≥

 =

0

0 1

−1



|t|p √

dt 1 − t2

π



|cos u|p du

40

M. Balazard

 ≥

1 −1

|t|1/r √

dt 1 − t2

one has p = p/r < 1/r

= I2 (s). Therefore, with .I = [−1, 1],

.



Qp ≥ 21−n I2 (s)1/p I1 (s)−1/p s = 21−n A(s)1/p

(1 < s < 2),

(11)

where A(s) = I2 (s)s/(s−1) I1 (s)−1/(s−1) .

.

Hence    1−

1/(s−1)  1 s 1 s 2s ( 2 ) (2 − s)   1 s (1 − 2s )2  32 − 2s

A(s) = 2

.

(1 < s < 2).

Putting .f (s) = ln (s), one has .

ln A(s) = ln 2+

sf (1 − 1/2s)+sf (1/2)+f (2 − s)−2f (1 − s/2)−sf (3/2−1/2s) · s−1

When s tends to 1, the last fraction tends to .

3 3 ln π + ψ(1/2) − ψ(1) = ln π − 3 ln 2, 2 2

with the usual notation .f =  /  = ψ. It follows that A(s) →

.

π 4

(s → 1).

Together with (11), this gives the inequality D ∗∗ (n, p, [−1, 1]) ≥ 21−n (π/4)1/p

.

and (3) now implies C(n, p) ≤ 22n−1 n!(8/π )1/p .

.

 We now prove an inequality involving three values of the function C.

On the Infimum of the Absolute Value of Successive Derivatives

41

Proposition 5.5 Let .p, q, r be positive real numbers such that 1 1 1 = + · p q r

.

Let m and n be positive integers. Then, .

C(m, q) C(n, r) C(m + n, p) ≥ · · (m + n)! m! n!

Proof Equivalently, by (3), one has to prove that D ∗∗ (m + n, p, I ) ≤ D ∗∗ (m, q, I ) · D ∗∗ (n, r, I ),

.

where I is a segment of the real line. In fact, if .P ∈ P m and .Q ∈ P n , then.P Q ∈ P m+n hence 

∗∗

|P (t)Q(t)|p dt

D (m + n, p, I ) ≤

.

p

I

≤



|P (t)|q dt

p/q   p/r |Q(t)|r dt ·

I

I

by the definition of .D ∗∗ (m + n, p, I ) and Hölder’s inequality. The greatest lower bound of the last term, when P runs over .P m and Q runs over .P n , is D ∗∗ (m, q, I )p · D ∗∗ (n, r, I )p .

.



The result follows.

5.6 An Open Question Finally, observing that

C(n, 2) ∼

.

2 2n · 2 n! π

(n → ∞),

(an exercise on Stirling’s formula from (8)), we ask the following question. Is it true that, for every .p > 0, the quantity .2−2n C(n, p)/n! tends to a limit when n tends to infinity?

42

M. Balazard

References 1. S.N. Bernstein, Sur l’ordre de la meilleure approximation des fonctions continues par des polynômes de degré donné. Mém. Cl. Sci. Acad. Roy. Belg. IV Fasc. 1, 1–104 (1912) 2. S.N. Bernstein, Extremal Properties of Polynomials and the Best Approximation of Continuous Functions of a Single Real Variable. Part I (G. R. O. L., Leningrad, Moscow, 1937). (in Russian) 3. G. Darboux, Mémoire sur les fonctions discontinues. Ann. de l’Éc. Norm. (2) 4, 57–112 (1875) 4. J. Dieudonné, Foundations of Modern Analysis, Pure and Applied Mathematics, vol. 10 (Academic Press, New York, 1969) 5. X. Huang, On extremal properties of algebraic polynomials, PhD Thesis, The Ohio State University, 1993 6. D. Jackson, On functions of closest approximation. Trans. Am. Math. Soc. 22, 117–128 (1921) 7. D. Jackson, Note on a class of polynomials of approximation. Trans. Am. Math. Soc. 22, 320– 326 (1921) 8. A. Korkine, G. Zolotareff, Sur un certain minimum. Nouv. Ann. 12, 337–356 (1873) 9. A. Kroó, F. Peherstorfer, On the zeros of polynomials of minimal Lp -norm. Proc. Am. Math. Soc. 101, 652–656 (1987) 10. A. Kroó, E. B. Saff, On polynomials of minimal Lq -deviation, 0 < q < 1. J. Lond. Math. Soc. II. Ser. 37, 182–192 (1988) 11. M.K. Kwong, A. Zettl, Norm Inequalities for Derivatives and Differences, Lecture Notes in Mathematics, vol. 1536 (Springer, 1992) 12. T.J. Rivlin, An Introduction to the Approximation of Functions (Dover Publications Inc., Mineola, 1981) 13. L. Schoenfeld, On the differentiability of indeterminate quotients. Math. Mag. 41, 152–155 (1968) 14. J. Soula, Sur une inégalité vérifiée par une fonction et sa dérivée d’ordre n. Mathematica 6, 86–88 (1932) 15. I.V. Tsenov, On a question of the approximation of functions by polynomials. Mat. Sb., Nov. Ser. 28, 473–478 (1951). (in Russian) 16. H. Whitney, Differentiability of the remainder term in Taylor’s formulaä. Duke Math. J. 10, 153–158 (1943) 17. E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, 4th edn. (Cambridge University Press, 1927) 18. J.W. Young, General theory of approximation by functions involving a given number of arbitrary parameters. Trans. Am. Math. Soc. 8, 331–344 (1907)

Friable Averages of Oscillating Arithmetic Functions Régis de la Bretèche and Gérald Tenenbaum

To the memory of Eduard Wirsing whose profound insights will continue fertilising our field.

Abstract We evaluate friable averages of arithmetic functions whose Dirichlet series is analytically close to some negative power of the Riemann zeta function. We obtain asymptotic expansions resembling those provided by the Selberg-Delange method in the non-friable case. An application is given to summing truncated versions of such functions. Keywords Riemann zeta function · Friable integers · Delay-differential equations · Selberg-Delange method

1 Introduction and Statements of Results Let .P + (n) denote the largest prime factor of an integer n, with the convention that + + .P (1) = 1. Given .y  1, an integer n is said to be y-friable if .P (n)  y. Let .S(x, y) stand for the set of y-friable integers not exceeding x and let us write traditionally .(x, y) := |S(x, y)|. Mean values of multiplicative functions over the set .S(x, y) attracted a lot of attention during the last decades—see, e.g., [3, 4, 6, 14– 16]. However few articles deal with oscillating summands. Alladi [1], Hildebrand [7, 8], and Tenenbaum [11] consider the case of the Möbius function. Hildebrand’s

R. de la Bretèche Université Paris Cité, Sorbonne Université, CNRS, Institut de mathématiques de Jussieu-Paris Rive Gauche, Paris, France e-mail: [email protected] G. Tenenbaum () Institut Élie Cartan, Université de Lorraine, Vandœuvre-lès-Nancy Cedex, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Maier et al. (eds.), Number Theory in Memory of Eduard Wirsing, https://doi.org/10.1007/978-3-031-31617-3_5

43

44

R. de la Bretèche and G. Tenenbaum

work [8] also deals with the function .eiϑ(n) , where .ϑ ∈ R and .(n) denotes the total number of prime factors, counted with multiplicity, of an integer n. In this work, we aim at handling a more general situation. For given .β > 0, .c ∈]0, 1[, .δ > 0, .κ > 0, such that .β + δ < 3/5, we consider the classes .E ± κ (β, c, δ) of those Dirichlet series .F (s) convergent for .σ = s > 1 and which may be represented on this half-plane in the form F (s) = ζ (s)±κ B (s)

.

(1.1)

 where the series .B (s) := n1 b(n)/ns may be holomorphically continued to a domain   + (1−δ−β)/(β+δ) .D (β, c, δ) := s ∈ C : σ > 1 − c/{1 + log |τ |} (1.2) with .σ := s, .τ := s, .log+ t = max(log t, 0) .(t > 0), and satisfies the conditions   s ∈ D (β, c, δ) ,

B (s)  {1 + |τ |}1−δ

.



B (s, y) :=

P + (n)y .

= B (s) + O



(1.3)

b(n) ns

1 Lβ+δ (y)



 y  2, σ > 1 − c/(log y)β+δ , , |τ |  Lβ+δ (y)

(1.4)

with r

Lr (y) := e(log y) (r > 0, y  2).

.

(1.5)

We then consider the class .H (κ, κ0 ; β, c, δ) of those arithmetic functions f whose associated Dirichlet series .F (s) belongs to .E − κ (β, c, δ) and furthermore possesses a majorant series .F † (s) = ζ (s)κ0 B † (s) in .E + κ0 (β, c, δ). We note that the approach implemented in this work can readily be extended to the case where the powers of the Riemann zeta function appearing in (1.1) are replaced by products of powers of Dedekind zeta functions—see [6] for details. Moreover, it is plain that all our results below generalise mutatis mutandis to the case when .κ is an arbitrary complex number. However we chose to focus on the real case in order to avoid some technicalities. We thus aim at establishing estimates for sums M(x, y; f ) :=



.

n∈S(x,y)

f (n)

Friable Averages of Oscillating Arithmetic Functions

45

of real arithmetic functions f with Dirichlet series F (s) :=

.

 f (n) ns

(s > 1)

n1

belonging to .H (κ, κ0 ; β, c, δ) for arbitrary values of the relevant parameters in the prescribed ranges. The quantities arising in our formal statements below depend on the theory of delay differential equations. We describe the behaviour of the relevant functions in Sect. 3. Let .hκ denote the unique continuous solution on .[0, ∞[ of the equation vhκ (v) = κhκ (v − 1)

(v > 1)

.

(1.6)

with initial condition hκ (v) = 1

.

(0  v  1).

(1.7)

Define further M(x; f ) := M(x, x; f ),

.

A(x, y; f ) := x

 M(y v ; f ) . (1.8) hκ (u − v)d yv −∞ ∞

Here and in the sequel, we write systematically .u := (log x)/ log y. Our first result furnishes a range in which .A(x, y; f ) is a good approximation to .M(x, y; f ). It involves a rapidly decreasing function .Rκ , precisely defined in Sect. 3.2 and satisfying (1.15) below. Theorem 1.1 Let .β > 0, .c > 0, .δ > 0, .κ > 0, .κ0 > 0, .β + δ < 3/5. Then, for suitable .ε = ε(β, δ) > 0 and uniformly for .f ∈ H (κ, κ0 ; β, c, δ) and .(x, y) in the range x  3,

.

exp{(log x)1−β }  y  x,

(Gβ )

we have

M(x, y; f ) = A(x, y; f ) + O

.

 xRκ (u) . Lε (y)

(1.9)

Let .ν := κ, .ϑ := κ = κ − ν. We aim at some asymptotic expansion for A(x, y; f ) in the spirit of results provided by the Selberg-Delange method—see [13, ch. II.5 & II.6]. However this is by no means a straightforward consequence of (1.8). The finer behaviour will be described in terms of the function .ϕκ , continuous

.

46

R. de la Bretèche and G. Tenenbaum

solution on .R+ , differentiable on .[1, ∞[, to the delay-differential equation vϕκ (v) + ϑϕκ (v) − κϕκ (v − 1) = 0,

(1.10)

.

with initial condition ϕκ (v) =

.

v −ϑ (1 − ϑ)

(0 < v  1).

(1.11)

It is worthwhile to observe right away that, if .κ = ν ∈ N∗ , then .ϕκ = hκ . It follows from above that .ϕκ is .C ∞ on .R  N and .C j on .]j, ∞[ for all .j  0. (j ) When .κ ∈ N∗ , the discontinuity of .ϕκ at .m ∈ [1, j ] is of the first kind. We may (j ) hence continue .ϕκ by right-continuity at m. We write   (j ) (j ) .δκ,m,j = ϕκ (m) − ϕκ (m − 0) κ ∈ N∗ , 1  m  j . (1.12) It will be seen that ψκ := ϕκ(ν)

(1.13)

.

occurs naturally in our study. A description of the asymptotic behaviour of .ψκ and its derivatives may be obtained from the general theory developed in [9]: we provide the necessary details in Sect. 3.2. For .v > 0, define .ξ(v) as the unique non zero real solution of the equation ξ 1 .e = 1 + vξ , and put .ξ(1) := 0. We have [13, lemma III.5.11] ξ(v) = log(v log v) + O

.

 log v 2 log v

(v  3),

and an asymptotic expansion may be derived through standard techniques. Define further .ζ0 (v) as the solution of the equation .eζ = 1 − vζ with largest negative imaginary part. By [9, lemma 1], we have ζ0 (v) = ξ(v) +

.

 1 π2 π ξ(v) + O − i ξ(v) − 1 2ξ(v)2 ξ(v)3

(v  2).

(1.14)

With this definition, we can state that the function .Rκ appearing in the statement of Theorem 1.1 and defined in Sect. 3.2 satisfies2 

v 1 .Rκ (v)  √ exp ζ0 (t/κ)dt (v  1). (1.15) − v κ

1 Here

and throughout we denote by .logk the k-fold iterated logarithm. and in the sequel we extend the meaning of the notation .f  g to .|f |  |g|, hence relevant to complex quantities.

2 Here

Friable Averages of Oscillating Arithmetic Functions

47

Let .dμf,y (t) be the measure on .R with Laplace transform μ f,y (s) :=

.

R

e−ts dμf,y (t) =

s 1−ϑ F (1 + s/ log y) s + log y

(σ > 0).

(1.16)

We then put, for .j  0, .y  2, Wj (v, y; f ) :=

.

∞

(t − v)j dμf,y (t)

(v  0),

(1.17)

v

(−1) X (x, y; f ) := ( − 1)! .



1/2

0

ϕκ() (u − t)W−1 (t, y; f )dt

(  1, y < x  y 

+1

(1.18)

).

It follows from estimates stated later—see in that order (5.1), (1.26), (5.15), and (5.10) infra—that Wj (v, y; f ) 

.

1 (log y)j +ϑ L

β+δ/2 (y

v)

(v  0, y  2).

(1.19)

Moreover, as will be shown in Sect. 5.2, writing .λ := log(x/y  ), X (x, y; f ) 

.

λ1−ϑ 1 1 +  · (1 + λ)(log y) (log y)+ϑ (log y)

(1.20)

Finally, for integer .J  0, real .y  3, put .εJ,y := {(2J + 2) log2 y}1/β / log y and consider the sets   .D J (y) := u  1 : min (u − j )  εJ,y . 1j min(u,J +1)

We can now state our second main result, where .{aj (f )}∞ j =0 is the sequence defined by the Taylor expansion .

 F (s + 1) = aj (f )s j κ s (s + 1) j 0

(|s| < c).

48

R. de la Bretèche and G. Tenenbaum

Theorem 1.2 Let β > 0, c > 0, δ > 0, κ0 > 0, κ > 0, ν := κ, β + δ < 3/5, J ∈ N.

.

Then, uniformly for f ∈ H (κ, κ0 ; β, c, δ), (x, y) ∈ Gβ , u = (log x)/ log y ∈ D J +ν (y),

.

we have A(x, y; f ) = x

.

  aj (f )ψκ(j +1) (u) xRκ (u)(log 2u)J +1 + O . (1.21) (log y)κ+j +1 (log y)κ+J +2

0j J

When .u ∈ D J +ν (y), . < u   + 1, and .κ ∈ N∗ , the above formula persists if the quantity .xUJ (x, y; f ) is added to the main term, with 

UJ (x, y; f ) :=

.

j J +ν+1

(−1)j +1 δκ,,j Wj (u − , y; f ). j!

(1.22)

If .u ∈ D J +ν (y), . < u   + 1, .  J + ν + 1, and .κ ∈ R∗  N∗ , formula (1.21) must be modified by restricting the summation to the (possibly empty) range .0  j   − ν − 2 and adding .xX (x, y; f ) to the main term. We note that .ψ1 = ω, Buchstab’s function—see, e.g., [13, § III.6.2]—, so we recover [11, th. 2], with some further precision, in the special case .f = μ, the Möbius function. Corollary 1.3 Let .β > 0, .c > 0, .δ > 0, .κ0 > 0, .κ > 0, .ν := κ, .β + δ < 3/5. Then, uniformly for .f ∈ H (κ, κ0 ; β, c, δ), .(x, y) ∈ Gβ , .u ∈ D ν (y), we have M(x, y; f ) =

.

  R (u) log(2u)  x κ  . B (1)ψ (u) + O κ log y (log y)κ+1

(1.23)

This describes the asymptotic behaviour unless .u  ν + 1 and .u  εν,y . When  < u   + 1, .  ν + 1, formula (1.23) must be modified according to the specifications described in the statement of Theorem 1.2. In particular, we have in all cases

.

M(x, y; f ) 

.

x (log y)min(κ+1,)

(y  < x  y +1 ).

(1.24)

Note that, for .u = 1, the Selberg-Delange method (see [13][ch. II.5]) furnishes, for .x  1, M(x; f ) =

.

 1  −(κ + 1) sin(π κ)x  (κ ∈ R+  N), . B (1)+O log 2x π(1 + log x)κ+1 (1.25)

Friable Averages of Oscillating Arithmetic Functions

M(x; f ) 

x Lβ (x)

49

(κ ∈ N∗ ).

(1.26)

Finally, we mention that the above results, the estimates of Sects. 3.2 and 3.3, and of [14], [3], open the way to upper bounds that are uniform for .x  y  2. The following corollary is proved in Sect. 7. We denote by .H ∗ (κ, κ; β, c, δ) the subclass of .H (κ, κ; β, c, δ) subject to the further condition that, for a suitable constant C, we have .B (s, y)  ζ (2ακ , y)C and .1/ζ (2ακ , y)C  B † (s, y)  ζ (2ακ , y)C when κ .s  ακ (x, y), defined as the saddle-point associated to .ζ (s, y) . Corollary 1.4 Let .β > 0, .c > 0, .δ > 0, .κ0 > 0, .κ > 0, .β + δ < 3/5, .1 < r < 3/2. Then there exists a constant .c0 such that uniformly for .f ∈ H ∗ (κ, κ; β, c, δ), we have 

−c0 u/(log 2u)2 1 e † (x  y  2), (1.27) + .M(x, y; f )  M(x, y; f ) Lr (y) (log y)κ+m−1 where .m := min(u, κ + 1).

2 Applications 2.1 Weighted Averages Consider the weighted analogue 

m(x, y; f ) :=

.

n∈S(x,y)

f (n) n

(2.1)

of .M(x, y; f ). In some situations, it is convenient to have an estimate for (2.1) at our disposal, parallel to that following from Theorems 1.1 and 1.2. This is the purpose of the following statement in which .μ∗f,y denotes the measure with Laplace transform .F (1 + s/ log y)/s ϑ and .{aj∗ (f )}∞ j =0 is the sequence of Taylor coefficients of .F (1 + s)/s κ . For the sake of further reference we note that aj∗ (f ) = aj (f ) + aj −1 (f )

.

(j  0),

with the convention that .ah (f ) = 0 if .h < 0. We also define ∞ 1 ∗ .Wj (v; f ) := (t − v)j dμ∗f,y (t)  , v )(log y)j +ϑ L (y β+δ/2 v

(2.2)

(2.3)

and, for . < u   + 1, X∗ (x, y; f ) :=

.

(−1) ( − 1)!



1/2 0

∗ ϕκ() (u − t)W−1 (t, y; f )dt.

(2.4)

50

R. de la Bretèche and G. Tenenbaum

Similarly to (1.20), writing .λ := log(x/y  ), we have X∗ (x, y; f ) 

.

1 1 λ1−ϑ +  ·  +ϑ (1 + λ)(log y) (log y) (log y)

(2.5)

Theorem 2.1 Let β > 0, c > 0, δ > 0, κ0 > 0, κ > 0, ν := κ, β + δ < 3/5, J ∈ N.

.

Then, uniformly for .f ∈ H (κ, κ0 ; β, c, δ), .(x, y) ∈ Gβ , .u ∈ D J +ν (y), we have m(x, y; f ) =

.

 aj∗ (f )ψκ(j ) (u) 0j J

(log y)κ+j

 Rκ (u)(log 2u)J . +O (log y)J +κ+1

(2.6)

When .u ∈ D J +ν (y), . < u   + 1, and .κ ∈ N∗ , the above formula persists provided the quantity .UJ∗ (x, y; f ) is added to the main term, with UJ∗ (x, y; f ) :=



(−1)j +1 δκ,,j ∗ Wj (u − ; f ), j!

.

j J +ν

(2.7)

where .δκ,m,j is defined in (1.12). If .u ∈ D J +ν (y), . < u   + 1, .  J + ν + 1, and .κ ∈ R∗  N∗ , formula (2.6) must be modified by restricting the summation to the (possibly empty) range ∗ .0  j   − ν − 2 and adding .X (x, y; f ) to the main term.  At the cost of a weakening of the error term, one can take advantage of the rapid decrease of the density of friable integers as u gets large in order to derive estimates valid without any restriction. For instance in the case of the Möbius function .μ, we have, uniformly for .x  y  2,  .

n∈S(x,y)

ω(u) μ(n) = n log y

1

x/y

m(t) dt + O t

where .ω denotes Buchstab’s function and .m(t) :=



 nt

1 (log y)2

μ(n)/n.

 ,

Friable Averages of Oscillating Arithmetic Functions

51

2.2 Truncated Multiplicative Functions In a recent preprint [2], Alladi and Goswami gave estimates for the summatory function  . (−k)ω(n,y) nx

 where .k ∈ N∗ , and .ω(n, y) := p | n, py 1. The results presented in Sect. 1 enable to consider more generally .M(x; fy ) for .f ∈ H (κ, κ0 ; β, c, δ) with .κ > 0 and .fy is defined as .fy (n) = f (m) if m is the largest y-friable divisor of n. Thus .fy is obtained from f by truncating its values at all large primes, similarly to the process classically used for additive functions. Some further notation is necessary to state our result. For integer .J  0 and constant .b > 0, we consider the sets  D J (b, y) := u  1 :

.

min

1j < min(u,J +1)

 (u − j ) > 1/(log y)b .

Our estimate for .M(x; fy ) is stated below for .u ∈ D J (b, y) and suitable b > 0. This produces an estimate in which the discontinuities of .ψκ+1 have no influence. Taking into account the contributions of the discontinuities described in Theorems 1.2 and 2.1 leads to a more complicated statement valid without restriction. We leave this to the reader.

.

Theorem 2.2 Let β > 0,

.

c > 0,

β + δ < 12 ,

δ > 0,

J ∈ N∗ ,

κ0 > 0,

κ > 0,

ν := κ,

b := (1 − 2β)/(1 − β).

Then, uniformly for .f ∈ H (κ, κ0 ; β, c, δ), .(x, y) ∈ Gβ , and .u ∈ D J +ν+1 (b, y), we have M(x; fy ) = x

.

(j )  aj (f )ψκ+1 (u) 0j J

(log y)κ+j +1

+O

 xRκ (u)(log 2u)J +1 . (log y)J +κ+2

(2.8)

In the special case .f (n) := (−k)ω(n) , where .ω(n) now counts without multiplicity the total number of prime factors of n, formula (2.8) represents a significant improvement over the corresponding estimate in [2]: it is valid in a much larger range since [2, th. 5.2] requires .β to be taken arbitrarily small, and it also furnishes an expansion according to negative powers of .log y whereas only the dominant term is provided in [2]. Note that for this particular function .a0 (f ) vanishes whenever .k = p + 1 for some prime p. This accounts for the dichotomy put forward in [2].

52

R. de la Bretèche and G. Tenenbaum

It is also noteworthy to remark that, under the assumptions of Theorem 2.2 and u  log2 y, an obvious modification of the proof of [5, th. 02] provides

.

a0 (f )e−γ (κ+1) x .M(x; fy ) = +O (log y)κ+1



x (log y)κ+2

 .

The main term agrees with that of (2.8) in view of (3.6) below. Hence (2.8) extends the scope of the above result by providing an expansion of the remainder term in a fairly large domain.

3 Solutions to Delay-Differential Equations 3.1 The Function hκ We continue .hκ on .R by setting .hκ (v) = 0 for .v < 0, so that (1.6) still holds for .v ∈ R  {0, 1}. From the general theory displayed in [9], we know—see [9, theorem 1]—that hκ (v)  v κ

(v > 1),

.

so the Laplace transform κ (s) := .h



∞

e−vs hκ (v)dv

0

converges for .σ := s > 0. Arguing as in [13, § III.6.3], which correa similar situation in the case .κ = 1, it is easy to show that sponds to κ (s)  = −κe−s hκ (s)/s, and then . sh hκ (s) =

.

1 s κ+1 (s)κ

(σ > 0),

(3.1)

where . denotes Dickman’s function, the solution of (1.6) when  s .κ is replaced by −1. We recall (see, e.g. [13, th. III.5.10]) that, writing .I (s) := 0 (et − 1)dt/t, we have

.

γ +I (−s)  . (s) = e

(s ∈ C).

(3.2)

Friable Averages of Oscillating Arithmetic Functions

53

By [9, eq. .(3.4 )], we have hκ (v) = AF (v; 0, −κ) + 2



.

An Fn (v; 0, −κ) + O(EN (v))

0n 0)

0

can be computed classically by showing from the delay-differential equation (1.10) that it satisfies a linear differential equation. We omit the details, which are similar to the computation of  . (s)—see, e.g., [13, th. III.5.10]. We obtain 1

ϕκ (s) =

.

s ν+1 (s)κ

(σ > 0),

(3.5)

and observe that the inverse Laplace integral ϕκ (v) =

.

1 2π i



evs

1+iR

s ν+1 (s)κ

ds

(v ∈ R)

2 converges for all .v = 0 since  . (s) = 1/s + O(1/s ) for .s = 1. Moreover, we recover the fact that .ϕκ (v) = 0 for .v < 0 by moving the integration line to the right at infinity. From (1.10), we see that .ψκ satisfies (3.3) with .(a, b) = (κ, −κ). By [9, th. 1], we have .F (v; κ, −κ) = 1 for the exceptional solution of this delaydifferential equation. Therefore, letting .δ0j denote Kronecker’s symbol, we get for .j  0 that

ψκ(j ) (v) = δ0j e−γ κ + 2



.

λj,n Fn (v; κ + j, −κ) + O(EN (v)), .

(3.6)

0n 1)

.

56

R. de la Bretèche and G. Tenenbaum

with initial condition .κ (v) = v κ−1 / (κ) for .0 < v  1, we also infer from [9] (see also [10]) that, for large v,   1   κ (v, ξ(v/κ) κ (v) = eγ κ + O v . v     1  ξ  (v/κ) exp γ κ − ξ(t/κ)dt , = 1+O v 2κπ κ   and so, in view of (1.14), (3.9) and (3.10), writing .H (v) := exp v/(log 2v)2 , Rκ (v) = κ (v) exp

.

 −π 2 v 2ξ(v)2

+O

 v  κ (v) = · 2 3 ξ(v) H (v)π /2+o(1)

(3.13)

Recall that, in the case .F (s) = ζ (s)κ B (s) ∈ E + κ (β, c, δ), we have, by [6, th. 1.1],   log 2u 1  + , (3.14) M(x, y; f ) = xκ (u)(log y)κ−1 B (1) + O log y (log y)κ

.

uniformly for .(x, y) ∈ Gβ , with .u := (log x)/ log y.

4 Proof of Theorem 1.1 4.1 Auxiliary Estimates Lemma 4.1 ([12]) For v  1, s = −ξ0 (v), we have    ζ0 (v) (s)|.  − ζ0 (v)   |s

.

(4.1)  

Proof The bound (4.1) coincides with [12, lemma 8].

Our second lemma states an estimate which may be proved similarly to [12, lemma 10], the details being left to the reader. We use the notation ακ = ακ (x, y) := 1 − ξ0 (u/κ)/ log y

.

ζ (s, y) :=



.

py

(1 − 1/ps )−1

(x  y  2),

(s > 0, y  2),

(4.2)

(4.3)

Friable Averages of Oscillating Arithmetic Functions

57

and, for b > 0, define the domain x  3, exp{(log2 x)b }  y  x

.

(Hb )

Lemma 4.2 Let b > 5/3. Uniformly for (x, y) ∈ (Hb ), we have 4

x ακ ζ (ακ , y)κ  xκ (u)eO(u/ξ(u) ) (log y)κ .

.

(4.4)

We next restate [13, lemma III.5.16] in the following weaker form. Lemma 4.3 Let 3/5 < b < 3/2. Then, for 0 < ε < 1/2 − b/3, we have   ζ (s, y) = ζ (s)sy (sy ) 1 + O

.

1  Lε (y)

(4.5)

with sy := (s − 1) log y and uniformly in the range y  2,

σ  1 − 1/(log y)2b/3+3ε/2 ,

.

|τ |  Lb (y).

Finally, we need an estimate for short sums of the coefficients of series in E+ κ (β, c, δ). Lemma 4.4 Let β > 0, c > 0, δ > 0, β + δ < 3/5, κ0 > 0, and let f † denote a non-negative arithmetic function with Dirichlet series in E + κ0 (β, c, δ). Then the estimate  x . f † (n)  z(log x)κ0 −1 + (4.6) Lβ+2δ/3 (x) x 0 be small, put .T := u2u Lβ+δ/2 (y), and let .{f † (n)}∞ n=1 denote the (β, c, δ). We first sequence of the coefficients of the majorant series .F † (s) ∈ E + κ

58

R. de la Bretèche and G. Tenenbaum

apply Perron’s formula (see, e.g., [13, th. II.2.3]) to get, for .(x, y) ∈ Gβ , M(x, y; f ) =

.

1 2π i



ακ +iT 2 ακ −iT 2

B (s, y)x s ds + R, sζ (s, y)κ

(4.8)

with R



.

P (n)y



x ακ f † (n) nακ (1 + T 2 | log(x/n)|)

x ακ ζ (ακ , y)κ + T

 |n−x|x/T

f † (n) 

xRκ (u) , Lβ+δ/2 (y)

by (4.4), (3.13), and (4.6). Next, we check that, for sufficiently small .δ, we have .T  Lr (y) for some .r < 3/2, and so apply formula (3.1) and Lemma 4.3 to derive, for .(x, y) ∈ (Gβ ), 2 .|τ |  T , .sy := (s − 1) log y, .L = Lε (y),    1  xeusy 1 + O 1/L2  xeusy sy hκ sy  xs 1+O 2 = . = · sζ (s, y)κ sζ (s)κ s{ζ (s)sy }κ (sy )κ L

(4.9)

We note that, when .κ ∈ N∗ , the singularity arising from the pole of .ζ (s) at .s = 1 is compensated by the corresponding zero of .sy , so that the main term is analytic on a zero-free region of the zeta function. Applying (4.1) and (3.11), we see that the contribution of the last error term to the integral of (4.8) is

.

 

x ακ L2



T2 0

{log(2 + τ )}κ dτ |ζ0 (u/κ) (−ζ0 (u/κ))|κ (1 + τ )

√ xRκ (u) u{log T }κ+1 xRκ (u) ·  2 L L

This is compatible with (1.9). Taking (1.4) into account, the same argument yields, for .x  y  2, M(x, y; f ) =

.

1 2π i



ακ +iT 2 ακ −iT 2

 xR (u) B (s)x s κ . ds +O sζ (s)κ syκ (sy )κ L

(4.10)

Considering (3.1) and since .

(s − 1)B (s) = sζ (s)κ



∞ 0

 M(ev ; f ) (σ > 1), e−v(s−1) d ev

(4.11)

Friable Averages of Oscillating Arithmetic Functions

59

we deduce from the convolution theorem that .sy B (s)hκ (sy )/{sζ (s)κ } is the Laplace transform of ∞   M(y v ; f ) t t −v d . .Jy (t) := e hκ log y yv 0 We plainly have .Jy (log x) = A(x, y; f ). Since .Jy (s) is holomorphic in any zero-free region of the zeta function, we may write 1 (4.12) .Jy (log x) = Jy (s)x s ds 2π i L where .L is the broken line .[b − i∞, b − iT 2 , ακ − iT 2 , ακ + iT 2 , b + iT 2 , b + i∞] with .b := 1 + 1/ log x. In order to prove (1.9), it therefore remains to show that the 2 2 contribution of the complement of the segment  .[α  κ − iT , ακ + iT ] is negligible. On the horizontal segments, we have .sy hκ sy  1 and, by standard bounds for the zeta function in the Vinogradov-Korobov region,  κ κ .F (s) = B (s)/ζ (s)  log(|τ | + 2) . The corresponding contribution is therefore . x(log T )κ /T 2 , which is plainly acceptable. Now, by [13, lemma III.5.12], we have sy hκ (sy ) = 1 + O

.

 1 + uξ(u) sy

(|τ | > T 2 ).

Thus the contribution of the vertical half-lines may be estimated by Perron’s formula for the main term .x s /{sζ (s)κ } and by the direct bound . x{1 + uξ(u)}/(τ 2 log y) for the remainder. This completes the proof of (1.9).

5 Proof of Theorem 1.2 5.1 The Case κ ∈ N∗ When .κ = ν ∈ N∗ , we have, by (1.16) and (4.11), μf,y (v) = M(y v ; f )/y v , . aj −κ−1 (f ) (−1)j ∞ j = v dμf,y (v) (log y)j j! 0

(5.1)

.

with the convention that .ah (f ) = 0 if .h < 0.

(j  0),

(5.2)

60

R. de la Bretèche and G. Tenenbaum

We may discard the case .u = 1 since the required formula is then a straightforward consequence of the Selberg–Delange method, as stated in [13, th. II.5.2]. When .u ∈ D J +ν (y), .u > 1, we apply (1.9) and (5.1) and write

u

.

ϕκ (u − v)dμf,y (v) = I1 + I2 + O

0



1



Lβ+δ/2 (x)

,

where the .Ij correspond respectively to the integration ranges .[0, 12 εy ], .] 12 εy , u− 12 ]; here and throughout, we write for simplicity .εy := εJ +ν,y . To evaluate .I1 we may use the fact that .ϕκ belongs the class .C J +κ+2 on 1 1 .[u − εy , u]. For .0  v  εy we hence have the Taylor-Lagrange expansion 2 2 

ϕκ (u − v) =

(j )

.

0j J +κ+1

(−v)j ϕκ (u) + R0 , j!

(5.3)

with (−1)J +κ+2 .R0 := (J + κ + 1)!



v

0

(v − t)J +κ+1 ψκ(J +2) (u − t)dt.

Therefore I1 =



(j )

.

0j J +κ+1

(−1)j ϕκ (u) j !(log y)κ+j +1



εy /2

v j dμf,y (v) + S0 ,

(5.4)

0

with S0 :=

.

(−1)J +κ+2 (J + κ + 1)!



εy /2 0

ψκ(J +2) (u − t)



εy /2

(v − t)J +κ+1 dμf,y (v)dt.

t

We first observe that, by (1.26), we may extend the integrals in (5.4) to infinity involving a remainder absorbable by that of (1.21). To this extent and in view of (5.2), the sum may be replaced by

.

 aj (f )ψκ(j ) (u) · (log y)κ+j +1

0j J

Next, noting that, for .j > ν, u  1, 0  v  u − 12 , ψκ(j ) (u − v)  Rκ (u − v)(log 2u)j −1  Rκ (u)(log 2u)j −1 evξ0 (u/κ) ,

.

(5.5)

which readily follows, as in [13, cor. III.5.15], from (3.8) and (3.11), we see that .S0 may be absorbed by the remainder of (1.21).

Friable Averages of Oscillating Arithmetic Functions

61

The integral .I2 can be handled as an error term. Indeed, it suffices to perform partial summation and apply (1.26) to see that .I2 does not exceed the error term of (1.21). This completes the proof of (1.21) when .u ∈ D J +ν (y) and .κ = ν ∈ N∗ . When .u ∈ D J +ν (y), .u > 1, we have . < u   + 1 for some integer . ∈ [1, J + ν + 1]. The Taylor-Lagrange formula must then take the contributions of the discontinuities into account. The quantity 

R1 :=

.

1j J +κ+1

(−1)j +1 j!



δκ,m,j (v + m − u)j

1mj u−v 0, .δ > 0, .κ0 > 0, .κ > 0, .β + δ < 3/5, f ∈ H (κ, κ0 ; β, c, δ). The integral

.

Zf,y (v) :=

.

(log y)1−ϑ 2π i

F (1 + s)y vs 1+iR

converges almost everywhere, the exceptional {v ∈ R : y v ∈ N}. Moreover, for any fixed .j  0,

s 1−ϑ ds s+1

set

being

(5.9) included

in

.



∞

.

t j Zf,y (t)dt 

v

1 , (log y)ϑ+j Lβ+δ/2 (y v )2

(5.10)

and, when .y v   1, .0  h  12 ,

v

.

Zf,y (t)dt  hϑ Lβ+δ/2 (y v / h log y)2ϑ−2 .

(5.11)

v−h

Proof The statement regarding convergence is implied by (5.6). Thus

v

.

0

Zf,y (w)dw =

1 2π i(log y)ϑ

F (1 + s) 1+iR

y vs − 1 ds. s ϑ (s + 1)

(5.12)

Friable Averages of Oscillating Arithmetic Functions

63

Splitting the integrand in (5.12) by isolating the term involving .y vs and moving the integration line for this part into a zero-free region of .ζ (1 + s), we get

v

.

0

−1 Zf,y (t)dt = 2π i(log y)ϑ

1+iR

 (log y)−ϑ F (1 + s) ds + O , s ϑ (s + 1) Lβ+δ/2 (y v )2

which implies (5.10) for .j = 0. The extension to .j  1 is immediate. The proof of (5.11) is more delicate and follows the approach of [6, lemma 3.4]. We observe at the outset that we may assume .h  1/Lβ+δ/2 (y v ) since (5.11) otherwise follows from (5.10). To simplify the exposition, we prove (5.11) in the case .y := e, and consequently assume .h  1/Lβ+δ/2 (ev ). We start with a general upper bound for bf (v) :=

.

1+iR

F (1 + s)evs ds, s ϑ (s + 1)

in which .ev  need not be bounded from below. Let .N ∈ N∗ be such that .ev  = |ev − N|. For .v  3, .n ∈ N∗ , let us specialize, in (5.7), .σ := 1/v, replace v by .v − log n, multiply out by .f (n)/n and sum up over ∗ .n ∈ N . We get bf (v) =

1/v+iT

.

1/v−iT

 1  F (1 + s)evs f † (n) ds + O . s ϑ (s + 1) Tϑ n1+1/v (1 + T |v − log n|) n1

By (4.6), we see that the contribution to the last sum of those integers n outside the interval .[N/2, 3N/2] is . v κ0 /T . That of the term .n = N is, still by (4.7), .



ev (ev

+ T ev )L

v 10 β+δ/2 (e )

·

Let us denote by V the complementary contribution, so that 

V 

.

0 N, this implies V 

.



1 T

 0mLβ+δ/2 (N )10

1 m+1

 mN |n−N |Lβ+δ/2 (N )10 (m+1)N

1 N (log N)β+κ0  · 10 T Lβ+δ/2 (N ) Lβ+δ/2 (ev T )4

f † (n)

64

R. de la Bretèche and G. Tenenbaum

If .T  N, we have similarly 

V 

.

0mT /2





0mT /2



1 (m + 1)N



f † (n)

mN/T 0 such that .ev−h   1 if .h  ce−v . In this case, the last term of the above upper bound is .



ev T 1+ϑ L

β1

(ev )10



1/ h · v 5 β1 (e / h)

T 1+ϑ L

If, to the contrary, .h > ce−v , then this last term is .



T ϑL

1 1  ϑ · v 10 T Lβ1 (ev / h)5 β1 (e )

Friable Averages of Oscillating Arithmetic Functions

65

Thus, in all cases, bf (b) − bf (v − h)  hT 1−ϑ (log T )κ +

.

+

v κ0 1 + ϑ 1+ϑ T T Lβ+δ/2 (ev T )4

1 + 1/(hT ) · T ϑ Lβ+δ/2 (ev / h)4

Selecting .T := 1/{hLβ+δ/2 (ev / h)2 }, we get (5.11) for .y = e and hence for general y on replacing v by .v log y and h by .h log y.   Lemma 5.3 Under the hypotheses of Lemma 5.2 and with the convention that ah (f ) = 0 if .h < 0, we have

.

(−1)j . j!



∞

Zf,y (v)v j dv =

0

aj −ν−1 (f ) (log y)j +ϑ

(j  0).

(5.13)

Proof The bound (5.10) guarantees via partial integration that the integrals (5.13) converge. We have s 1−ϑ F (1 + s/ log y) , Z f,y (s) = s + log y

(5.14)

.

and observe that this implies, for .ϑ = 0, dμf,y (v) = Zf,y (v)dv.

(5.15)

.

From (5.14), we get, for .|s| < c log y, .J  1,  s J +2+ν  aj (f )s j +ν+1 s ν+1 F (1 + s/ log y) = + O Z J f,y (s) = κ s (1 + s/ log y) log y (log y)κ+j +1 (log y)κ+J +2

.

=

0j J

∞

Zf,y (v)e−vs dv =

0

+OJ

 0j J +ν+1



(−1)j s j j!



∞

Zf,y (v)v j dv

0

s J +2+ν . (log y)κ+J +2  

This is all we need. We are now in a position to complete the proof of (1.21) when .κ ∈ view of (4.7), we may plainly assume that .x = y u ∈ 12 + N∗ . Since, from (5.14) and (3.5), .

R+  N.

sy B (s) 1 B (s) = = Z κ (sy ) log y. f,y (sy )ϕ sζ (s)κ syκ (sy )κ sζ (s)κ syϑ syν+1 (sy )κ

In

66

R. de la Bretèche and G. Tenenbaum

we deduce by (4.12) that A(x, y; f ) = Jy (log x) = x(Zf,y ∗ ϕκ )(u).

.

(5.16)

The above convolution may be estimated in much the same way as in the case κ ∈ N∗ . We write u . ϕκ (u − v)Zf,y (v)dv = I1 + I2 + I3 , (5.17)

.

0

where .I1 corresponds to the contribution of .v ∈ [0, 12 εy ], .I2 to that of 1 1 1 .v ∈] εy , u − ], and .I3 to that of .[u − , u]. 2 2 2 Partial integration yields .I2  1/Lβ+δ/2 (y) in view of (5.10). To bound .I3 , we use (5.11) and (1.11). Consequently  I3  (u − v)

.

 

u

−ϑ 

v

u  Zf,y (t)dt 

+

u−1/2

0

1/2

dh hLβ+δ/2 (y u / h)2−2ϑ

1 · Lβ+δ/2 (y u )1−ϑ



To estimate .I1 , we argue differently according to whether .u ∈ D J +ν (y) or not. In the first instance, the Taylor expansion (5.3) is still valid and we derive the required conclusion using (5.10), (5.13) and (5.5). In the complementary case, let . ∈ [1, J + ν + 1] be defined by . < u   + 1. Writing the Taylor expansion at order . provides ϕκ (u − v) =

.

v  (−v)j ϕκ(j ) (u) (−1) + (v − t)−1 ϕκ() (u − t) dt. j! ( − 1)! 0

0j ν + 2, .β < 3/5, .(x, y) ∈ Gβ , we appeal to (1.9), (1.21) and (3.13) to get the upper bound e−c1 u/(log 2u) , (log y)2κ 2

.



valid for any .c1 < π 2 /2. When .u > (log y)β/(1−β) , we redefine .ακ as the saddle point associated to the Perron integral for .M(x, y; τκ ), hence involving .ζ (s, y)κ . We then have, by Perron’s formula, arguing as in [11, lemma 2], for .r < 3/2, M(x, y; f ) =

.

1 2π i



ακ +iLr (y)2 ακ −iLr (y)2



B (s, y)x s xκ (u) −c2 u . + x ds + O (u)e κ sζ (s, y)κ Lr (y)

We conclude following the proof of [11, th. 1] by appealing to the bound [11, (2.4)] for .ζ (s, y)/ζ (ακ , y) and to the saddle point estimate stated in [3, § 2.1] for .M(x, y; τκ ), which extends to .M(x, y, f † ).

8 Proof of Theorem 2.2 For f as in the statement, we put .g = f ∗ μ, so that .g ∈ H (κ + 1, κ0 + 1; β, c, δ). We may plainly assume y, and hence x, sufficiently large throughout the proof.

Friable Averages of Oscillating Arithmetic Functions

69

Let z be a parameter to be defined later. By the hyperbola principle, we have 

M(x; fy ) =

g(n)

.

x 

n∈S(x/z,y)

n

+



M

x , y; g − M , y; g z d z

x

dz

=: S1 + S2 − S3 . We can replace .x/n by .x/n in .S1 with an error not exceeding that of (2.8) granted that .

xκ+1 (u)(log y)κ xRκ+1 (u) ·  z (log y)J +1

This is certainly the case for the choice .z := eu (log y)κ+J +2 . Under the assumptions of the statement, we then have .z = y o(1) and we may apply Theorem 2.1 to g, getting the suitable approximation S1 ≈ x



.

(j )

aj∗ (g)

0j J

ψκ+1 (u − uz ) (log y)κ+j +1

·

Here and in the sequel of this proof we use the symbol .A ≈ B to indicate that two quantities A and B agree to within an error not exceeding that of (2.8). By Theorem 1.1 and (5.16), we have, writing .ut := (log t)/ log y, S2 ≈

.

0

1/2

 ϕκ+1 (u − v − ud ) dμg,y (v). d

(8.1)

dz

The inner sum may be rewritten as

z

.

ϕκ+1 (u − v − ut )

1

dt = V21 + V22 , t

with

uz

V21 := (log y)

.

0

ϕκ+1 (u − v − w)dw,

z

V22 := − 1

ϕκ+1 (u − v − ut )

dt · t

70

R. de la Bretèche and G. Tenenbaum

The contribution of .V21 to (8.1) is .



≈ (log y)

0j J +ν+2

=

(−1)j j!



uz 0



(j )

ϕκ+1 (u − w)dw

(j ) (j )  aj (g){ψκ+1 (u) − ψκ+1 (u − uz )}

(log y)κ+j +1

0j J

∞

v j dμg,y (v)

0

,

where Lemma 5.3 and (5.11) have been used. The quantity .V22 is handled by writing 

V22 ≈

.

0hJ +ν+2



≈

(−1)h+1 (h) ϕκ+1 (u − v) h!



z

uht

1

dt t

(h)

γh+1 ϕκ+1 (u − v) (log y)h

0hJ +ν+2

,

where .{γh }∞ h=0 is the sequence of Taylor coefficients of .sζ (1 + s) at the origin. Carrying back into (8.1), we obtain the contribution 

γh+1 (log y)h

.

0hJ +ν+2



≈

 0j J +ν+2



j!

v j dμg,y (v)

0

(log y)h+j +ν+2+ϑ

(h+j +1)  γh+1 aj (g)ψκ+1 (u)

0hJ 0j J

(log y)κ+2+h+j

(j +1)  ψκ+1 (u){aj +1 (f ) − aj +1 (g)}

≈

∞

(h+j +1)  γh+1 aj (g)ψκ+1 (u)

0hJ 0j J

≈



(h+j )

(−1)j ϕκ+1 (u)

(log y)κ+2+j

0j J

·

Finally, we have S3 ≈ x

.

(j +1)  aj (g)ψκ+1 (u − uz ) 0j J

(log y)κ+j +2

·

Gathering our estimates and using the fact that .a0 (g) = a0 (f ), we obtain (2.8) as required.

Friable Averages of Oscillating Arithmetic Functions

71

References 1. K. Alladi, Asymptotic estimates for sums involving the Möbius function II. Trans. Amer. Math. Soc. 272, 87–105 (1982) 2. K. Alladi, A. Goswami, Parity results concerning the generalized divisor function involving small prime factors of integers. Preprint. December (2021) 3. S. Drappeau, Remarques sur les moyennes des fonctions de Piltz sur les entiers friables. Q. J. Math. (Oxford) 67(4), 507–517 (2016) 4. E. Fouvry, G. Tenenbaum, Entiers sans grand facteur premier en progressions arithmétiques. Proc. London Math. Soc. (3) 63, 449–494 (1991) 5. R.R. Hall, G. Tenenbaum, Divisors, Cambridge Tracts in Mathematics, no. 90 (Cambridge University Press, 1988) 6. G. Hanrot, G. Tenenbaum, J. Wu, Moyennes de certaines fonctions multiplicatives sur les entiers friables, 2. Proc. Lond. Math. Soc. (3) 96, 107–135 (2008) 7. A. Hildebrand, On a problem of Erd˝os and Alladi. Monat. Math. 97, 119–124 (1984) 8. A. Hildebrand, On the number of prime factors of integers without large prime divisors. J. Number Theory 25, 81–86 (1987) 9. A. Hildebrand, G. Tenenbaum, On a class of difference differential equations arising in number theory. J. d’Analyse 61, 145–179 (1993) 10. H. Smida, Sur les puissances de convolution de la fonction de Dickman. Acta Arith. 59(2), 124–143 (1991) 11. G. Tenenbaum, Sur un problème d’Erd˝os et Alladi, in Séminaire de Théorie des Nombres, Paris 1988–89, ed. by C. Goldstein, Prog. Math., vol. 91 (Birkhäuser, 1990), pp. 221–239 12. G. Tenenbaum, Crible d’Ératosthène et modèle de Kubilius, in Number Theory in Progress, ed. by K. Gy˝ory, H. Iwaniec, J. Urbanowicz, Proceedings of the Conference in Honor of Andrzej Schinzel, Zakopane, Poland 1997 (Walter de Gruyter, Berlin, 1999), pp. 1099–1129 13. G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3rd edn., Graduate Studies in Mathematics 163 (American Mathematical Society, 2015) 14. G. Tenenbaum, J. Wu, Moyennes de certaines fonctions multiplicatives sur les entiers friables. J. Reine Angew. Math. 564, 119–166 (2003) 15. G. Tenenbaum, J. Wu, Moyennes de certaines fonctions multiplicatives sur les entiers friables, 3. Compos. Math. 144(2), 339–376 (2008) 16. G. Tenenbaum, J. Wu, Moyennes de certaines fonctions multiplicatives sur les entiers friables, 4. Actes du colloque de Montréal, 2006, Centre de Recherches Mathématiques, CRM Proceedings and Lecture Notes 46, 129–141 (2008)

Ein quaternäres Waring-Goldbach-Problem Jörg Brüdern

Eduard Wirsing zum Gedenken. Seine Mathematik stirbt nie.

Abstract If the Riemann hypothesis is true for all Dirichlet L-functions, then all large even natural numbers are the sum of a prime, a square of a prime and two cubes of primes. Keywords Waring-Goldbach problem · Circle method · Additive representations

1 Die Frage Prachar [10] hat bei gegebenem .n ∈ N für die Anzahl der Lösungen der diophantischen Gleichung p1 + p22 + p33 + p44 + p55 = n

.

(1)

in Primzahlen .p1 , . . . , p5 eine asymptotische Formel gefunden und konnte folgern, daß es für hinreichend große ungerade n stets prime Lösungen von (1) gibt. In diesem ästhetisch attraktiven Resultat sind ganz gewiß einige Variable redundant. So ist die Aufgabe, die Gleichung (1) mit .p5 = 3 zu lösen, gelegentlich als Pracharsches Problem bezeichnet worden. Zur Einordnung in das weite Feld der additiven Primzahltheorie lohnt ein kurzer Blick auf das allgemeine Waring-Goldbach-Problem. Hier ist zu gegebenen ganzen

J. Brüdern () Universität Göttingen, Mathematisches Institut, Göttingen, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Maier et al. (eds.), Number Theory in Memory of Eduard Wirsing, https://doi.org/10.1007/978-3-031-31617-3_6

73

74

J. Brüdern

Exponenten .kj mit .ks ≥ ks−1 ≥ . . . ≥ k1 ≥ 1 die in Primzahlen zu lösende Gleichung p1k1 + p2k2 + · · · + psks = n

.

(2)

vorgelegt. Bei solchen Fragen wirkt im Hintergrund aller zur Zeit verfügbaren Methoden ein Konvexitätsprinzip, das Angriffe grundsätzlich abwehrt, wenn nicht mindestens die Ungleichung

.

s  1 >2 kj

(3)

j =1

erfüllt ist. Bei binären Problemen .(s = 2) besteht deshalb zur Zeit wenig Hoffnung, obwohl bei richtiger Interpretation in der Gleichung (1) aller Voraussicht nach nicht nur die fünfte, sondern auch noch die vierte und dritte Potenz verzichtbar sind, denn nach den von Hardy und Littlewood im dritten Teil ihrer berühmten Reihe ‘Partitio Numerorum’ vorgestellten Prinzipien sollten der Lösbarkeit von .p1 + p22 = n nur algebraische Hindernisse im Wege stehen: n darf kein Quadrat sein und hat gewissen Kongruenzbedingungen modulo 6 zu genügen. Bei den ternären Problemen .(s = 3) hat (3) nur die Lösungen k1 = k2 = 1,

.

k3 ≥ 1,

und in allen diesen Fällen haben Vinogradov (.k3 = 1, z.B. [2, §6.4]) und Hua (.k3 ≥ 2, [9]) für die Anzahl der primen Lösungen von (2) eine asymptotische Formel angegeben. Die ternäre Klasse ist also unter der Bedingung (3) befriedigend bearbeitet. Bei .s = 4 sind deshalb Fälle mit .k1 = k2 = 1 ohne Interesse, und wegen (3) verbleiben dann noch k1 = 1,

.

k2 = k3 = 2,

k4 ≥ 2

(4)

3 ≤ k4 ≤ 5.

(5)

und k1 = 1,

.

k2 = 2,

k3 = 3,

Die in (4) beschriebene Familie hat Hua [9] ebenfalls erledigt, doch die drei Fälle in (5) sind offen. Das Pracharsche Problem reiht sich hier als der Fall .k4 = 4 ein. Trotz mannigfacher Verbesserungen der von Vinogradov entwickelten Techniken hat es bei den drei in (5) beschriebenen Problemen in den letzten achtzig Jahren keinerlei Fortschritte gegeben. In dieser Situation ist es vielleicht gerechtfertigt, den Status der aufgeworfenen Frage unter plausiblen Hypothesen hinsichtlich der Verteilung der Primzahlen in arithmetischen Progressionen zu untersuchen. Als Kronzeugen für ein solches Vorgehen können Hardy und Littlewood [7] benannt werden, die das ternäre Goldbach-Problem nur behandeln konnten, wenn die

Ein quaternäres Waring-Goldbach-Problem

75

komplexen Nullstellen aller Dirichletschen L-Funktionen sämtlich in der Halbebene Re z ≤ 12 liegen. Diese erweiterte Riemannsche Vermutung bezeichnen Hardy und Littlewood mit .R ∗ , was hier gerne übernommen wird. Auch Hooley [8] hat in seiner wichtigen, die Summe von einer Primzahl und zwei Quadraten betreffenden Arbeit ∗ .R postulieren müssen. In beiden Fällen hat es nicht lange gedauert, bis die beiden Ergebnisse auch ohne die Hypothese .R ∗ bestätigt wurden. In dieser Note soll der Fall .k4 = 3 von (5) unter Annahme von .R ∗ behandelt werden. Für das Pracharsche Problem reichen die hier vorgestellten Methoden nicht aus. .

Satz 1.1 Ist .R ∗ wahr und n eine hinreichend große gerade Zahl, dann gibt es Primzahlen .p1 , p2 , p3 , p4 mit p1 + p22 + p33 + p43 = n.

.

(6)

Notation Wie üblich bezeichnet .μ die Möbiussche Funktion, .φ den Eulerschen Totienten und . die von Mangoldtsche Funktion. Für ganze Zahlen .a, b ist .(a, b) der größte gemeinsame Teiler. Der von den meisten Autoren gemiedene größte gemeinsame Teiler von 0 und 0 kommt hier gelegentlich vor und ist als größer als jede natürliche Zahl anzusehen. Das Zeichen p ist für Primzahlen reserviert. Ferner wird die weit verbreitete Abkürzung .e(α) = exp(2π iα) verwendet. Ab § 4 treten Aussagen auf, die einen Parameter .ε enthalten. Jede dieser Aussagen ist wahr, sobald dem Symbol .ε eine positive reelle Zahl zugewiesen wird. Die in den üblichen Vinogradovschen und Landauschen Symbolen versteckten Konstanten hängen dann von der Konkretisierung von .ε ab. Diese Vereinbarung ermöglicht es, von .A  nε und .B  nε auf .AB  nε zu schließen.

2 Ein Mittelwertsatz In diesem Abschnitt wird ein Mittelwertsatz bereitgestellt, der auch bei verwandten Fragen nützlich sein sollte. Das noch zu formulierende Ergebnis betrifft die Verteilung der Primzahlen im quadratischen Mittel, kodiert wird das Resultat jedoch in der Sprache von trigonometrischen Polynomen. Ein solches ist gegeben durch F (α) =



.

(m)e(αm).

m≤n

Ist .α nahe bei .a/q mit teilerfremden .a ∈ Z, .q ∈ N, dann wird .μ(q)φ(q)−1 F (α − a/q) eine gute Approximation von .F (α) sein. Hier soll der dabei entstehende Fehler im quadratischen Mittel kontrolliert werden. Für eine präzise Formulierung sei 1≤Q≤

.

1√ 2 n

(7)

76

J. Brüdern

ein noch freier Parameter. Die zu Paaren ganzer Zahlen .a, q mit .0 ≤ a ≤ q ≤ Q und .(a, q) = 1 gebildeten Intervalle M(q, a) = {α ∈ [0, 1] : |qα − a| ≤ Q/n}

.

sind dann disjunkt. Deren Vereinigung sei mit .M und wenn nötig genauer mit .M(Q) bezeichnet. Für .α ∈ M(q, a) wird F ∗ (α) = μ(q)φ(q)−1 F (α − a/q)

.

gesetzt. Das definiert eine Funktion .F ∗ : M → C. Jetzt kann die zentrale Abschätzung leicht formuliert werden. Satz 2.1 Für Q gelte (7). Ist .R ∗ wahr, dann auch die Ungleichung 

|F (α) − F ∗ (α)|2 dα  Q2 (log n)6 .

.

M

Der nachfolgende Beweis dieses Satzes stützt sich auf wesentliche Vorarbeiten von Goldston und Vaughan [6], dort vor allem §5, und benutzt Methoden der multiplikativen Zahlentheorie. Zwei vorbereitende Hilfssätze sind Varianten von Abschätzungen, die implizit in [6, §5] zu finden sind. Der erste davon betrifft das zu einem Dirichlet-Charakter .χ gebildete trigonometrische Polynom Fχ (α) =



χ (m)(m)e(αm).

.

m≤n

Für .δ > 0 sei  Iχ (δ) =

δ

.

−δ

|Fχ (α)|2 dα.

(8)

Lemma √2.2 Sei .χ ein primitiver Dirichlet-Charakter modulo q. Ferner sei .q ≤ n und . n < δ −1 ≤ n. Ist die Riemannsche Vermutung für die Dirichletsche LFunktion .L(s, χ ) wahr, dann gilt auch Iχ (δ)  δ 2 n2 (log n)4 .

.

Beweis Zur Abkürzung sei . = (2δ)−1 . Mit einer bekannten Technik von Gallagher [4, Lemma 1] ergibt sich zunächst  Iχ (δ)  δ 2

∞

.

−∞

  

 1≤m≤n t 0 and .|f (x)|/g(x) < B holds for some positive constant B and all sufficiently large positive values of x.) It is easy to see that the truth of the PNT implies that .ζ (1 + it) = 0 for all real t. On the other hand, N. Wiener’s Tauberian theory established the converse relation; thus the PNT can be considered as “equivalent” to the nonvanishing of .ζ (1 + it). The belief arose that the only path to the PNT was via .ζ (s) and complex (or Fourier) analysis. The mathematical world was astonished when, in the middle of the twentieth century, elementary proofs of the PNT were discovered. The first of these were based on a formula discovered by A. Selberg that can be written  .

(n) log n +

n≤x



(m) (n) = 2x log x + O(x)

(1.3)

mn≤x

or, equivalently, ψ(x) +

.

 x  1  . ψ(x/n) (n) = 2x + O log x n≤x log x

(1.4)

This formula is quite easy to prove today (see below); the real achievements were realizing that it might lead to a new proof of the PNT and carrying this out. Indeed, Selberg and P. Erd˝os were each able to derive proofs of the PNT from the formula without reference to .ζ (s) or complex analysis.

Wirsing’s Elementary Proofs of the Prime Number Theorem with Remainder Terms

101

Remarks on (1.4) 1. Mertens’ elementary approximation  (n) .

n≤x

n

= log x + O(1)

(1.5)

shows that .

1  (n) ψ(x/n) log x n≤x n x/n

is an average of the values of .ψ(y)/y. Thus (1.4) connects .ψ(x), which counts primes or prime powers, with a weighted convolution sum involving two primes. A Tauberian argument is needed to pry these terms apart. 2. Selberg’s formula yields the following upper estimate for the number of primes in a relatively short interval. Such an inequality was not accessible to Chebyshev, and it will be useful to us in the sequel. Lemma 1.1 If .0 < h ≤ x, then ψ(x + h) − ψ(x) ≤ 2h + O(x/ log x) .

.

This follows from the first version of the Selberg formula with arguments .x + h and x. Subtracting and dropping the (nonnegative!) double sum yields  .

(n) log n ≤ 2(x + h) log(x + h) − 2x log x + O(x),

x 1, for the error term in the Selberg formula (1.4) is itself of size .x log−1 x. Wirsing [7] overcame this problem by successively improving this error term. At about the same time, another elementary proof of (1.6) with arbitrary .α was established by E. Bombieri [2] by using higher order analogues of the Selberg formula.

102

H. G. Diamond

Theorem 1.3 Formula (1.6) is valid for any positive value of .α. We are going to survey Wirsing’s method of establishing these two PNT error terms, describing his main ideas and motivations and giving some key details. The principal sources for this article are Wirsing’s two papers [6, 7] and the author’s survey article [3].

2 The Selberg Formula Mechanism Here we describe the setup for the Selberg formula. Beyond its own interest, this discussion helps motivate the improved version that will be described in Sect. 4. We begin with some notation. Define .∗, the multiplicative convolution of arithmetic functions .f, g, by (f ∗ g)(n) :=



.

f (i)g(j ) =

ij =n



f (i)g(n/i);

i|n

we write this simply as .f ∗ g(n). The summatory function satisfies  .

f ∗ g(n) =



f (i)g(j ) =

ij ≤x

n≤x





f (i)

i≤x

g(j ).

j ≤x/i

Define three special arithmetic functions, .1, e and .μ, by .1(n) = 1 for all n; e(1) = 1 and .e(n) = 0 for .n ≥ 2; and .μ is the Möbius function. e is the unity element for .∗, and .μ is the convolution inverse of 1:

.

f ∗ e for all f, and 1 ∗ μ = e.

.

Here are two other useful operators on arithmetic functions. Define .L by Lf (n) = f (n) log n and .L2 f (n) = f (n) log2 n. .L is a derivation:

.

L(f ∗ g) = f ∗ Lg + (Lf ) ∗ g.

.

Chebyshev’s identity (1.1) can be expressed in these terms as  ∗ 1 = L1

.

(2.1)

or . = L1 ∗ μ. Also, define the “.1/n” operator .T by .Tf (n) = f (n)/n. We have T(f ∗ g) = Tf ∗ Tg. For example, .T 1 ∗ T μ = T (1 ∗ μ) = T e = e. Now we complete the proof of Selberg’s formula. First apply .L to (2.1) and then convolve each side by .μ and use the earlier identities to obtain

.

L +  ∗  = L2 1 ∗ μ.

.

(2.2)

Wirsing’s Elementary Proofs of the Prime Number Theorem with Remainder Terms

103

Summing this yields the left side of (1.3); for the right side, note that  .

L2 1(n) = x log2 x − 2x log x + 2x + O(log2 x).

n≤x

Taking .f = 2L1 + c1 1 + c2 e with suitable constants .c1 , c2 , we find 

√ f ∗ 1(n) = x log2 x − 2x log x + 2x + O( x log x),

.

n≤x

and so  .

L(n) +  ∗ (n) =

n≤x



f ∗ 1 ∗ μ(n) + R(x) = 2x log x + O(x),

n≤x

√ since (with . x log x  x 3/4 )        2 .|R(x)| :=  {(L 1 − f ∗ 1) ∗ μ}(n) ≤ O((x/n)3/4 ) |μ(n)| = O(x).   n≤x

n≤x

The version of Selberg’s formula to be given in Sect. 4 improves this bound by using estimates of the summatory function of .μ.

3 Error Estimates, I Sketch of Wirsing’s α = 3/4 Proof The argument is delicate, so we describe the main innovations in some detail. Wirsing begins with a form of Selberg’s formula, to which he applies an iteration procedure. A key step is a Tauberian argument based on an ingenious non-euclidean version of the isoperimetric inequality. The starting point is the formula  .

n≤x

rn log x =



rm rn + O(1),

(3.1)

mn≤x

with rn := (1 − (n))/n, n ≥ 2, and r1 := 1 − 2γ ,

.

(3.2)

where γ denotes Euler’s constant. This follows by combining Selberg’s formula, identity (1.1), Dirichlet’s divisor result, and summation by parts. The raison d’être

104

H. G. Diamond

for 2γ in r1 is that this occurs in the asymptotic formula 1 (1 − (n)) → 2γ as x → ∞, n n≤x

.

which is equivalent to the PNT [1, Th. 5.9]. (The last fact is not used in the proof; insertion of the 2γ serves to simplify the form of (3.1).) Also, Lemma 1.1 and similar estimates give the near-continuity bound        .  ≤ log x + O 1 r . n   y log y

(3.3)

y 0 and all ξ ≥ 0. From these formulas the estimate σ (ξ ) = O(ξ −α ) will be derived, which in turn easily leads to (1.6). If we naively assume |σ (t)| < δ holds for some δ > 0 and all sufficiently large t and simply apply (3.1 ) and (3.3 ), we obtain only |σ (t)| < δ + o(1), so we have made no progress at all. (The symbol o(1) denotes a function that tends to 0 as the argument goes to infinity.) The “secret sauce” for effectively estimating σ (x) is a Tauberian argument based on the following geometric inequality.

Wirsing’s Elementary Proofs of the Prime Number Theorem with Remainder Terms

105

Lemma 3.1 Let R be the plane rectangle {(u, v) : |u| ≤ A, |v| ≤ B}, and C a polygonal arc lying inside R and represented by continuous piecewise linear functions u(t), v(t) for t1 ≤ t ≤ t2 . Let   F := 

t2

.

t1

  u(t) dv(t)

and  L :=

t2

.

max(|u (t)|, |v (t)|) dt,

t1

that is, L is the length of C in the metric in which each section of the figure is measured by the larger of its projections upon the coordinate axes. Then F ≤ (L + 2A + 2B)

.

2AB . √ A + B + A2 + B 2

(3.4)

The lemma is proved under the initial assumption that the figure is a polygon, and a kind of isoperimetric inequality is established: the area is estimated from above in terms of this non-euclidean measure of perimeter. Then it is shown how to remove possible double points from the original arc and close up the figure to make a polygon. The “cost” of the last operations is having L + 2A + 2B in (3.4) in place of L. The theorem, in turn, is established by an induction, using the following parameters. Suppose ξ is sufficiently large, K ≥ 1 a constant, and α ∈ (0, 1). Also, a small number is fixed. (The interdependency of these quantities is specified below.) It is shown inductively that if |σ (η)| ≤ Kη−α

(3.5)

.

for some K ≥ 1 and all η ≤ ξ , then this inequality holds with the same K and α for all η > 0. The integration interval [0, ξ ] of (3.1 ) is broken into a union of n subintervals (with n suitably chosen), and Lemma 3.1 is applied to each of the integrals, with u(t) := σ (ξ − t) and v(t) := σ (t). Denoting a generic subinterval ((ν − 1)ξ/n, νξ/n] by iν , we write       .Fν :=  σ (ξ − η) dσ (η) ,

 Lν :=

iν

Aν := max |σ (ξ − η)|,

.

η∈iν

max(|σ (ξ − η)|, |σ (η)|) dη ,

iν

Bν := max |σ (η)| . η∈iν

106

H. G. Diamond

Now rewrite (3.4) for the integral over iν as 

−1 −2 −2 −1 Fν ≤ 2(Lν + 2Aν + 2Bν ) A−1 + B + A + B , ν ν ν ν

.

(3.6)

and insert the estimates Aν ≤ K

.

 n − ν −α , ξ n

Bν ≤ K

 ν − 1 −α , ξ n

Aν , Bν ≤ K1 ,

and  Lν ≤

(1 + ) dη = (1 + ) ξ/n.

.

iν

We find, for sufficiently large ξ , Lν + 2Aν + 2Bν ≤ (1 + )ξ/n + 4K1 ≤ (1 + )2 ξ/n.

.

Also, Bν−1 + A−1 + Bν−2 + A−2 ν ν    ξ α  ν − 1 α  n − ν  α ν − 1 2α  n − ν 2α + + + ≥ . K n n n n

.

Let   −1 2  ν − 1 α  n − ν α ν − 1 2α  n − ν 2α .Gν (α) := + + + . n n n n n With these preparations, we return to our main estimate. Recalling (3.1 ) and noting that O(log ξ )/ξ < /ξ α for fixed and sufficiently large ξ , we have |σ (ξ )| ≤

.

n n  1 Fν + ξ −α ≤ (1 + )2 Kξ −α Gν (α) + K ξ −α . ξ ν=1

Note that

ν

ν=1

Gν (α) is a Riemann sum for 

J (α) :=

.

1



−1  2 x α + (1 − x)α + x 2α + (1 − x)2α dx.

0

Wirsing shows that J (3/4) < 1 by a clever use of both the arithmetic-geometric mean inequality and Hölder’s inequality to prove that the integrand of J is smaller than 1. To save space, we simply quote the result of a numerical integration:

Wirsing’s Elementary Proofs of the Prime Number Theorem with Remainder Terms

107

J (3/4) ≈ 0.983. (Also, the exponent of the theorem could not be pushed to 4/5, because J (4/5) ≈ 1.007 > 1.) With α = 3/4, choose > 0 such that (1 + )3 J (3/4) + 2 ≤ 1.

.

With now fixed, next suppose n is chosen so large that n  .

Gν (3/4) ≤ (1 + )J (3/4).

1

With these parameters, we have |σ (ξ )| ≤ Kξ −3/4 {(1 + )3 J (3/4) + } ≤ (1 − )Kξ −3/4 .

.

Since σ is continuous, (3.5) holds for some ξ > ξ . Thus there is no maximal ξ for   which this inequality holds, and so (3.5) is valid for all arguments.

4 Error Estimates, II Sketch of Wirsing’s Proof for All Powers of the Logarithm The argument is recursive: the PNT error term established at a given stage is used to make an improved version of Selberg’s formula. A new fundamental inequality for convolutions of real functions (of independent interest!) provides the mechanism for carrying out the needed Tauberian argument. This PNT estimate, in turn, is applied to further improve the Selberg formula, etc.   We begin by stating the new inequality. Lemma 4.1 Let f and g be real valued Lebesgue measurable functions on [0, ∞) and suppose 1 . lim x→∞ x

 0

x

1 f (y) dy =: F, lim x→∞ x



2

x

g 2 (y) dy =: G.

(4.1)

0

Also, for x > 0, let h(x) :=

.

1 x



x

f (x − y)g(y) dy

(4.2)

0

and suppose 1 . lim x→∞ x

 0

x

h(y) dy = 0.

(4.3)

108

H. G. Diamond

Then, for arbitrary x0 > 0, 1 x→∞ x

.



x

lim

h2 (y) dy ≤

x0

1 F G. 2

(4.4)

Remarks 1. The essential point of the lemma is the factor 1/2; Schwarz’s inequality gives (4.4) without this factor. 2. The conditions of Lebesgue measurability and x0 > 0 are included for generality; in our case the functions are bounded and piecewise continuous, and we can take x0 = 0.

x 3. Instead of (4.3), one could assume 0 f (y) dy = o(x). 4. The example f (x) = g(x) = sin x shows the result optimal. The proof of this lemma is quite long, and we content ourselves with an analogous result for which we can give a simple proof (by a completely different method). The lim sup conditions on f, g, and h define seminorms on the Besicovitch space B 2 of almost periodic functions (although f, g, and h of this lemma need not be in this class). Here is an estimate of a similar type for functions that are periodic. Lemma 4.2 Let f and g be real Lebesgue measurable functions, each of period 1, and suppose 

1

.

 f (y) dy =: F, 2

0

1

g 2 (y) dy =: G.

0

Also, let 

1

h(x) :=

.

f (x − y)g(y) dy,

0

and assume 

1

.

h(y) dy = 0.

0

Then 

1

.

0

h2 (y) dy ≤

1 F G. 2

Proof of Lemma 4.2 f and g are integrable on [0, 1] by Schwarz’s inequality, and they have Fourier coefficients fˆ(n) =



1

.

0

f (y)e−2π iny dy, g(n) ˆ =



1 0

g(y)e−2π iny dy.

Wirsing’s Elementary Proofs of the Prime Number Theorem with Remainder Terms

109

Also,  0=

1

.

h(x) dx =

 1

0

0

1





1

f (x − y) g(y) dy dx =

1

f (x) dx

0

0

g(y) dy. 0

Thus at least one of g(0), ˆ fˆ(0) is 0; WLOG say fˆ(0) = 0. Since f is real, |fˆ(−n)| = |fˆ(n)| for all n = 0. Now by Parseval’s identity and the last two observations 

1

F =

.

f 2 (y) dy =

0

∞ 

|fˆ(n)|2 = 2

∞ 

n=−∞

|fˆ(n)|2 .

1

ˆ Thus, trivially, |fˆ(n)|2 ≤ F /2 for each n = 0. Also, |h(n)| = |fˆ(n)g(n)|, ˆ and by Parseval again, 

1

.

h2 (y) dy =

0



2 ˆ |h(n)| =

n

 n=0

1 F  2 2 |g(n)| ˆ ≤ F G. |fˆ(n) g(n)| ˆ ≤ 2 2 n=0

 



Akin to Remarks 3. and 4. of Lemma 4.1, the condition h = 0 can be replaced here by fˆ(0) = 0; also the example f (x) = g(x) = sin 2π x shows the result is optimal. We introduce some more notation and explain a formula we shall presently use. Analogous to functions from §3, for x < 1 set r(x) = r ∗ (x) = 0, and set r(x) :=

.

 1 − (n) n≤x

n

− 2γ , r ∗ (x) := log x −

 (n) n≤x

n

− γ,

x ≥ 1.

Since r ∗ (x) = r(x) + O(1/x), we can switch freely between the functions. Chebyshev’s identity (2.1) has the following equivalent but less familiar variant: Lμ = − ∗ μ. To see this, first note that L(1 ∗ μ) = Le = 0. Now 0 = μ ∗ L(1 ∗ μ) = μ ∗ L1 ∗ μ + μ ∗ 1 ∗ Lμ =  ∗ μ + Lμ.

.

We are going to use the “T” version of the variant formula: .

 (i)μ(j ) 1 1 μ(n) log n = − ( ∗ μ)(n) = − . n n ij

(4.5)

ij =n

We now show how a PNT error term leads to corresponding estimates of m(x) := n≤x μ(n)/n . It is elementary that m(x) = O(1); also, the PNT implies

110

H. G. Diamond

that m(x) = o(1); we shall use an elaboration of the latter proof in our argument. As in the previous section, we use the notation ξ := log x. Lemma 4.3 If r(x) = O(ξ −α ) for some α > 0, then m(x) = O(ξ −α ). Proof The argument will again be inductive. By assumption there is a constant R > 1 such that |r ∗ (x)| ≤ Rξ −α for x > 1. Let > 0 be chosen such that < (1 − )α /12. Choose M ≥ 8R −α and so large, initially dependent upon x(!), that |m(y)| ≤ M log−α y, for all y ≤ x.

.

Using the definition of r ∗ and (4.5) we have 

r∗

.

n≤x

 x  μ(n) n

n

=

 μ(n)  n

n≤x

log

  (i)μ(n) x −γ − = (ξ − γ ) m(x). n in in≤x

Write n r ∗ (x/n) μ(n)/n as a Stieltjes integral and apply the Dirichlet hyperbola method: let x1 = x 1− , x2 = x , break the integral at x1 , and apply integration by parts to the portion extending over (x1 , x]. We find  (ξ − γ ) m(x) =

x1

.

r∗

1−

x  y

(ξ − γ ) |m(x)| ≤ R log−α x2

 dm(y) +

x2

1−



x1

.

 x   − m(x1 ) dr ∗ (y) , m y

|dm(y)| + 2M log−α x1

1−



x2

|dr ∗ (y)| .

1−

Using the simple estimates 

x1

.

1−

 x2  1  (n) ≤ 2 log x1 , + γ ≤ 3 log x2 , |dm| ≤ |dr ∗ | ≤ log x2 + n n 1− n≤x n≤x 1

2

valid for all large x, we have (ξ − γ ) |m(x)| ≤ 2R log−α x2 log x1 + 6M log−α x1 log x2

.

= {2R −α (1 − ) + 6M(1 − )−a } log1−α x ≤ (3/4)Mξ 1−α , and so |m(x)| ≤ (4/5)Mξ −α . For x < y ≤ x + 1, and large x, we have |m(y)| ≤ |m(x)| +

.

4 1 1 ≤ M log−α x + ≤ M log−α y. x 5 x

Thus the claimed estimate holds in the range (x, x + 1] with the same M, provided only that we started with a sufficiently large number x. The argument can be

Wirsing’s Elementary Proofs of the Prime Number Theorem with Remainder Terms

111

repeated indefinitely with this M, so |m(y)| ≤ M log−α y for all y > 1.

.

  Remark The procedure for selecting M in this lemma is not effective. Therefore, this proof cannot establish the PNT with an error term whose O-constant is uniform with respect to the exponent α. Next, we show how a PNT error term yields a Selberg-type formula with an improved error term. Recall rn := (1 − (n))/n, n ≥ 2, and r1 := 1 − 2γ .

.

Lemma 4.4 If r(x) :=



n≤x rn

= O(ξ −α ) for some α > 0, then



ξ r(x) =

(3.2 )

.

rm rn + O(ξ −α log3 ξ ).

(4.6)

mn≤x

Proof We shall establish the related formula .

 {T(n) log n + (T ∗ T)(n)} = ξ 2 + c ξ + c + O(ξ −α log3 ξ ),

(4.7)

n≤x

for appropriate constants c, c . It is elementary to pass from  to the r form, as we noted after (3.2). The “1/n” operator T is introduced here for use in the conclusion of Wirsing’s argument. Our proof of (4.7) is based on the identity (cf. (2.2)) TL + T ∗ T = TL2 1 ∗ Tμ.

.

2 As described in Sect. 2, we seek a function F so that n≤x TL 1(n) is approximated by an expression of the form n≤x F (x/n)/n. Convolving this with Tμ will yield F (x) as our main term; the error term will be handled separately. We have  .

TL2 1 =

n≤x

 log2 n n≤x

n

=

 log2 x  log3 x +c+O 3 x

for some constant c. Take F (x) = log2 x + c1 log x + c2 .

.

(4.8)

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H. G. Diamond

(To make a smaller error term, we took F to be a continuous function rather than the summatory function of an arithmetic function.) A small calculation shows, for suitable constants c1 , c2 , that .

1 1 log3 x F (x/n) = + c + O n 3 x n≤x

(4.9)

with some constant c . By adding and subtracting the F term, we obtain .

 {TL2 1 ∗ Tμ}(n) = F (x) + Z(x), n≤x

where the error term is Z(x) :=

.

   log2 j 1  x  μ(k) − F . j j jk k k≤x j ≤x/k

We estimate Z(x) using the hyperbola method: Take x1 := xξ −α−2 and x2 := and write Z = Z1 + Z2 with

ξ α+2 ,

Z1 (x) :=

.

 log2 (x/k)  μ(k)    log2 j 1  x  μ(k)   − F = c +O j j jk k x/k k

k≤x1 j ≤x/k

k≤x1

for some constant c , by (4.8) and (4.9). Now c



.

μ(k)/k = c m(x1 )  log−α x1  log−α x

k≤x1

by Lemma 4.3. Also, .

 log2 (x/k) |μ(k)| x1 1  log2 (x/k) ≤ ≤ log2 x  log−α x . x/k k x x k≤x1

k≤x1

For Z2 we first sum on k: Z2 (x) =





.

j ≤x2 x1 β − 1; thus 2α ≥ α > βN − 1, and    2  ξ/2 σN (ξ − η) σN (η) dη + O(ξ −βN+1 ) .|σN (ξ )| ≤ ξ 0   ξ/2 1/2  ξ/2 1/2 2 2 ≤ σN 2 (η) dη σN2 (ξ − η) dη + O(ξ −βN+1 ) ξ 0 ξ 0  1/2 ≤ SN + o(1) sup |σN (η)| + O(ξ −βN+1 ). ξ/2≤η≤ξ

For sufficiently large ξ , by the choice of N,  1/2 SN + o(1) ≤ 2−1−β < 2−1−β N+1 ,

.

and so we have |σN (ξ )| ≤ 2−1−β N+1

.

sup

|σN (η)| + Aξ −βN+1

ξ/2≤η≤ξ

with a suitable constant A. We now choose B so large that B ≥ 4A and |σN (η)| ≤ Bη−β N+1

.

(4.18)

holds for ξ/2 < η < ξ . It then follows that |σN (ξ )| ≤

.

1 −β N+1 3 Bξ + Aξ −β N+1 ≤ Bξ −β N+1 . 2 4

Since σN is continuous, (4.18) holds on an interval (ξ/2, ξ1 ] with ξ1 > ξ , and so the set of points for which this inequality is valid has no upper bound. It follows that (4.18) holds for all η from some point onward. Combining (4.18) with (4.15) we conclude that ρ(ξ ) = σN (ξ ) + O(ξ −βN )  ξ −β N+1 .

.

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H. G. Diamond

Now, as α → α1 − (again, α1 is given after (4.11)), βN +1 → α2 := α1 +

.

β − α1 > α1 . (β + 1)N +1

Thus ρ(ξ )  ξ −α holds for arbitrarily large exponents α. Further Elementary PNT Estimates Using generalizations of the Selberg formulas with weights, the author and J. Steinig and others have given elementary proofs of the PNT with error terms of the type (1.2) (for details and references see [3, 4]). However, the exponents all have been much smaller than de la Vallée Poussin’s α = 1/2, and some 50 years have passed since the appearance of the last results of this kind. While there has been progress in some areas that are largely elementary, such as sieves and combinatorial number theory, analytic methods have had greater success recently, e.g. the work of Y. Zhang, et al. Elementary methods in prime number theory may have to await the arrival of their Riemann for a further major advance.

References 1. P.T. Bateman, H.G. Diamond, Analytic Number Theory. An Introductory Course (World Scientific, Singapore, 2004). Reprinted, with minor changes, in Monographs in Number Theory, vol. 1, 2009. MR2111739 2. E. Bombieri, Sulle formule di A. Selberg generalizzate per classi di funzioni aritmetiche e le applicazioni al problema del resto nel “Primzahlsatz”. Riv. Mat. Univ. Parma (2) 3, 393–440 (1962). MR0154860 3. H.G. Diamond, Elementary methods in the study of the distribution of prime numbers. Bull. Amer. Math. Soc. (N.S.) 7(3), 553–589 (1982). MR0670132 4. A.F. Lavrik, Methods of studying the law of distribution of primes (Russian), in International Conference on Analytic Methods in Number Theory and Analysis (Moscow, 1981). Trudy Mat. Inst. Steklov., vol. 163 (1984), pp. 118–142. MR0769880. Translation: Proc. Steklov Inst. Math. 1985 (American Math. Soc., Providence, 1985), pp. 141–167 5. H.L. Montgomery, R.C. Vaughn, Multiplicative Number Theory I: Classical Theory, Cambridge Studies in Advanced Mathematics, vol. 97 (University Press, Cambridge, 2007). MR2378655 6. E. Wirsing, Elementare Beweise des Primzahlsatzes mit Restglied. I. J. Reine Angew. Math. 211, 205–214 (1962). MR0150116 7. E. Wirsing, Elementare Beweise des Primzahlsatzes mit Restglied. II. J. Reine Angew. Math. 214/215, 1–18 (1964). MR0166180

Diophantine Analysis Around [1, 2, 3, . . . ]

.

Carsten Elsner and Christopher Robin Havens

Dedicated to the memory of Professor Eduard Wirsing (1931–2022)

Abstract The transcendence of the regular infinite continued fraction .z := [1, 2, 3, 4, 5, . . . ] was first proven by C. L. Siegel in 1929. The value of .z is a ratio of the values of modified Bessel functions. In this paper our diophantine analysis around .z takes its starting point with its rational convergents and deals with an asymptotic approximation formula for .z and with the construction of a sequence of quadratically irrational approximations using these convergents. Finally, we study various error sums for .z which are also defined by the rational convergents. Keywords Continued fractions · Error sums · Recurrences · Bessel functions

1 Introduction of the Zopf-Number In this paper we study a special number whose partial denominators form one of the simplest arithmetic sequences, namely .1, 2, 3, . . . . Indeed, in 1929 Siegel [8, 9] laid the groundwork for our study by treating the subject of our work as a special case among a family of quasi-periodic continued fractions as a ratio of modified Bessel functions of the first kind, .

∞  Ia/b (2/b) = a + kb k=0 , Ia/b+1 (2/b)

C. Elsner () Institute of Computer Sciences, FHDW University of Applied Sciences, Hannover, Germany e-mail: [email protected] C. R. Havens PMP Prison Mathematics Project, Phoenix, AZ, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Maier et al. (eds.), Number Theory in Memory of Eduard Wirsing, https://doi.org/10.1007/978-3-031-31617-3_9

119

120

C. Elsner and C. R. Havens

where .a, b ∈ Z with .b > 0 and .a + b > 0. With the method later named after him, values of analytic functions which satisfy a linear differential equation and whose coefficients fulfill analytic and algebraic conditions in their Taylor expansion can be proved to be transcendental. In this paper we restrict our analytic and diophantine investigations only to the special continued fraction 1

z := 1 +

.

= [1, 2, 3, 4, 5, . . . ] .

1

2+ 3+

1 4 +.

..

In Sect. 2, we establish an asymptotic estimation around the error of approximation of .z by its rational convergents .pn /qn . We then introduce a new concept, called the quadratic convergents of an irrational number .ξ . Let .ξ = [a0 , a1 , a2 , . . . ] be an irrational number where .a0 ≥ 1. For .n ≥ 1, the quadratic convergents are given by   Q n (ξ ) := a0 , a1 , . . . , an−1 ,

.

with characteristic polynomial .qn−1 X2 − (pn−1 − qn−2 )X − pn−2 = 0. Our results in Sect. 3 constitute very few of the myriad interesting properties of the quadratic convergents for the continued fraction .z = [ k ]∞ k=1 , for which we will henceforth refer to as the “Zopf-number”. The name, translating to “twist”, has been adopted by the authors because of the twist of .N around unity after taking .G n := gcd(qn−1 , pn−1 − qn−2 , pn−2 ) through positive values of n, and for the way its linear and quadratic convergents weave around the real line at .z. Specifically,  .

Gn

∞ n=1

= 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, . . . .

We then move into Sect. 4 and establish results on error sums of the Zopf-number in terms of Bessel functions.

2 Rational Convergents of the Zopf-Number Theorem 2.1 Let pn /qn be the n-th convergent of the Zopf-number z. Then we have  ln ln(qn ) pn   . z −  ∼ 2 qn qn ln(qn )

.

Diophantine Analysis Around .[1, 2, 3, . . . ]

121

Proof By setting z = [a0 , a1 , a2 , . . . ], we have an = n + 1 for n ≥ 0. Then by the well known inequalities for convergents of an irrational number, we have .

 1 1 pn  1 1  ≤ z − ≤ . <  <  2 qn (qn + qn+1 ) qn qn qn+1 (n + 4)qn (n + 2)qn2

(1)

From the recursion formula for the denominators qn of the convergents, we now get the following inequalities: (n + 2)qn ≤ qn+1 ≤ (n + 3)qn

.

(n ≥ 0) ,

(2)

since (n + 2)qn = an+1 qn ≤ qn+1 ≤ an+1 qn + qn = (1 + an+1 )qn = (n + 3)qn .

.

Using the induction principle and (2), lower and upper bounds for qn depending only on n can be proved: (n + 1)! ≤ qn ≤ (n + 2)!

.

(n ≥ 0) .

(3)

For n = 0 the inequalities in (3) are true because of 1 ≤ q0 = 1 ≤ 2. Next, let (3) be true for some integer n ≥ 0. Using both (2) and the induction hypothesis twice, we obtain on the one side, (n + 2)! = (n + 2)(n + 1)! ≤ (n + 2)qn ≤ qn+1 ,

.

and on the other side, qn+1 ≤ (n + 3)qn ≤ (n + 3)(n + 2)! = (n + 3)! .

.

Thus (3) is shown. From Stirling’s formula we have the asymptotic relation .

ln(n!) ∼ n ln(n) ,

(4)

which follows from [1, Eq. 6.1.41]. From (3), if we first take logarithms and then multiply on both ends by 1 = ln(n!)/ ln(n!), we have  .

1+

 ln(n + 1)(n + 2)  ln(n + 1)  ln(n!) ≤ ln(qn ) ≤ 1 + ln(n!) ln(n!) ln(n!)

(n ≥ 2) . (5)

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C. Elsner and C. R. Havens

Here, both fractions inside the parentheses tend to zero when n increases, which follows from (4). Combining (4) and (5), we find two functions ε1 (n) and ε2 (n), such that .

lim εk (n) = 1

n→∞

(k = 1, 2) ,

(6)

and ε1 (n)n ln(n) ≤ ln(qn ) ≤ ε2 (n)n ln(n)

.

(n ≥ 2) .

(7)

We now take logarithms of the inequalities in (7): .





ln ε1 (n) + ln(n) + ln ln(n) ≤ ln ln(qn ) ≤ ln ε2 (n) + ln(n) + ln ln(n) (n ≥ 3) . (8)

In the next step, we combine the inequalities in (7) and (8) to form: .

ε1 (n) ln(n) ln(qn ) ε2 (n) ln(n)



≤ (n ≥ 3). ≤ n ln ln(qn ) ln ε2 (n) + ln(n) + ln ln(n) ln ε1 (n) + ln(n) + ln ln(n)

Taking (6) into account, both the left and right-hand fraction tend to 1 for increasing n. This proves .

ln(qn ) ∼ 1. n ln ln(qn )

(9)

Finally, we access the inequalities in (1) and rearrange them in an equivalent form: .

ln(qn ) n q 2 ln(qn )  n ln(qn ) pn  · < n · z − ·  < n ln ln(qn ) n + 4 ln ln(qn ) qn n ln ln(qn ) n + 2

(n ≥ 3) .

By (9) we conclude that .

lim

n→∞

 q 2 ln(q )  pn   n  n · z −  = 1 ln ln(qn ) qn

⇐⇒

 ln ln(qn ) pn   . z −  ∼ 2 qn qn ln(qn )

This completes the proof of the theorem. For the sequence of the numbers qn , see A001053 in OEIS.



Diophantine Analysis Around .[1, 2, 3, . . . ]

123

3 Quadratic Convergents of the Zopf-Number Let ξ := [a0 , a1 , a2 , . . . ]

.

be an irrational number greater than 1, such that .a0 ≥ 1. Again, we denote the (rational) convergents of .ξ by .pm /qm for .m ≥ 0. Then, 1 .

2 (2 + am+1 )qm

 1 pm   < ξ −  < 2 qm am+1 qm

(m ≥ 0) .

(10)

Both inequalities in (10) result from inequalities (8) and (12) in [5, §13] and from the recurrence formula .qm+1 = am+1 qm + qm−1 . Let the quadratic convergents of .ξ be given by Q n (ξ ) := [ a0 , a1 , . . . , an−1 ]

.

for .n ≥ 1. Proposition 3.1 We have for every integer .n ≥ 1, .

  max 0,

1  1 1  1 1 − − , · 2 2 + an a0 2 + a0 an qn−1   1 1 1 < ξ − Q n (ξ ) < + · 2 . a0 an qn−1

(11)

Proof We have Q n (ξ ) = [ a0 , a1 , . . . , an−1 , Q n (ξ ) ] .

.

Therefore, the rationals .

p0 p 1 pn−1 , ,... , q0 q1 qn−1

are convergents of .ξ as well as convergents of .Q n (ξ ). Hence, we obtain four inequalities,  1 pn−1  1  < ξ − ,.  < 2 2 qn−1 (2 + an )qn−1 an qn−1

(12)

 1 1 pn−1   < Q (ξ ) − .  <  n 2 2 qn−1 (2 + a0 )qn−1 a0 qn−1

(13)

.

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C. Elsner and C. R. Havens

We now link these inequalities twice with the triangle inequalities. On the one side, we have     pn−1   pn−1   =  ξ− + .ξ − Q n (ξ ) − Q n (ξ )  qn−1 qn−1  pn−1   pn−1   ≤ ξ −  +  Q n (ξ ) −  qn−1 qn−1 (12),(13)


max − , − 2 2 2 + an a0 qn−1 2 + a0 an qn−1  1 1 1   1 1  , · 2 . = max − − (15) 2 + an a0 2 + a0 an qn−1 

(14) and (15) complete the proof of the Proposition.

3.1 An Application to the Zopf-Number Proposition 3.2 We have for every integer n ≥ 1 with |a0 − an | ≥ 3,    ξ − Q n (ξ ) > max

.

1 1 1 , · 2 . (2 + a0 )(3 + a0 ) (2 + an )(3 + an ) qn−1

(16)

Corollary 3.1 For the Zopf-number z we have the inequalities .

  5 1 <  z − Q n (z) < 2 2 12qn−1 4qn−1

(n ≥ 3) .

(17)

Proof of Corollary 3.1 The number z is such that an = n + 1, and so |a0 − an | = n. The lower bound in (17) then follows directly from (16). The upper bound in (17)

Diophantine Analysis Around .[1, 2, 3, . . . ]

125

is a consequence of the right-hand inequality in (11). The latter is because of n ≥ 3 and 1 1 n+2 5 + = ≤ . a0 an n+1 4

.



This proves the corollary. Proof of Proposition 3.2 We assume the condition |a0 − an | ≥ 3. Case 1 Let a0 − an ≥ 3. Then, we obtain a0 ≥ 3 + an ⇐⇒ −

.

⇐⇒

1 1 ≥ − a0 3 + an

(18)

1 1 1 1 2 + an 1 1 − ≥ − · = · . 2 + an a0 2 + an 2 + an 3 + an 2 + an 3 + an

Case 2 Let an − a0 ≥ 3. Interchanging a0 and an in Case 1, we obtain from (18), .

1 1 1 1 − ≥ · . 2 + a0 an 2 + a0 3 + a0

(19)

Finally, (16) follows from (18), (19), and from the left inequality in (11) of

 Proposition 3.1. Proposition 3.3 Let n ≥ 1 with an ≥ 2 + a0 . Then we have

ξ − Q n (ξ )

.

> 0 for n ≡ 0 (mod 2) , < 0 for n ≡ 1 (mod 2) .

(20)

Proof pn−1 /qn−1 is a common rational convergent of ξ and Q n (ξ ). Depending on whether n is odd or even, we obtain from (10) with m = n − 1 and a0 = an , (−1)n−1 Q n (ξ ) + (−1)n

.

pn−1 1 > . 2 qn−1 (2 + a0 )qn−1

(21)

Similarly, from (10) we obtain  1 pn−1   . < ξ − 2 qn−1 an qn−1

.

(22)

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C. Elsner and C. R. Havens

Setting Q(n) := (−1)n−1 Q n (ξ ) + (−1)n ξ , this gives  pn−1  pn−1 + (−1)n ξ − qn−1 qn−1   pn−1  pn−1  ≥ (−1)n−1 Q n (ξ ) + (−1)n − ξ −  qn−1 qn−1

Q(n) = (−1)n−1 Q n (ξ ) + (−1)n

.

1

(21),(22)

>

2 (2 + a0 )qn−1

−

1 2 an qn−1

≥

1 2 (2 + a0 )qn−1

−

1 2 (2 + a0 )qn−1

= 0.

This proves the inequalities in (20) and completes the proof of the proposition.



Theorem 3.2 The quadratic convergents Q n (z) of the Zopf-number z satisfy the inequalities Q 2 (z) < Q 4 (z) < Q 6 (z) < . . . < z < . . . < Q 7 (z) < Q 5 (z) < Q 3 (z) < Q 1 (z), (23)

.

where .

lim Q n (z) = z .

(24)

n→∞

Proof Equation (24) follows immediately from Corollary 3.1, so it remains to prove (23). From the regular continued fraction expansion of the Zopf-number, we have that an = n + 1 for n ≥ 0. Thus, an = n + 1 ≥ 3 = 2 + a0 is fulfilled for n ≥ 2. Then, Proposition 3.3 yields Q 2m (z) < z < Q 2m+1 (z)

.

(m ≥ 1) .

(25)

Next, we prove that     z − Q n+1 (z) < z − Q n (z)

.

(n ≥ 3) .

For this purpose, note that 4 ≤ n + 1 = an < [ an , an−1 , . . . , a1 ] =

.

2 2 , or Consequently, we have qn2 > 16qn−1 > 15qn−1

.

5 1 < 2 2 4qn 12qn−1

(n ≥ 3) .

qn . qn−1

(26)

Diophantine Analysis Around .[1, 2, 3, . . . ]

127

Finally, we apply Corollary 3.1 twice, namely for n + 1 and for n. This gives   z − Q n+1 (z)
qn−1 − 5 n 5 n (31)

pn−1 − qn−2 ≥ pn−1 −

.

(33)

Diophantine Analysis Around .[1, 2, 3, . . . ]

129

Additionally we have (31)

(30)

qn−1 ≥ nqn−2 >

.

2 npn−2 ≥ pn−2 . 3

(34)

The assertion of the lemma for .n ≥ 3 follows from (33) and (34). Also for .n = 1 and .n = 2 the statement is correct, because .



H P1 = H X2 − X − 1 = 1 = 1 − 0 = p0 − q−1 ,

2

H P2 = H 2X − 2X − 1 = 2 = 3 − 1 = p1 − q0 . 

The lemma is proven. Lemma 3.4 We have for all even integers .n ≥ 2, n

G n = gcd qn−1 , pn−1 − qn−2 , pn−2 ≡ 0 mod . 2

.

(35)

But it seems that much more is true. Conjecture 3.1 We have for all .n ≥ 1,

Gn =

.

1 if n ≡ 1 (mod 2) , n/2 if n ≡ 0 (mod 2) .

We will prove Lemma 3.4 below in Sect. 3.4. Now, from Lemmas 3.3 and 3.4 we have for every even integer .n ≥ 2,

2zqn−1 H Q n (z)  , n

.

(36)

since the difference .pn−1 − qn−2 in (32) has a simple asymptotic expansion pn−1 − qn−2 = qn−1 ·

.

p

n−1

qn−1

−

qn−2  ∼ zqn−1 , qn−1

and, consequently, .pn−1 − qn−2  zqn−1 . Note that we have the limits .

pn−1 = z n→∞ qn−1 lim

and .

lim

n→∞

qn−2 = lim [ 0, n, n − 1, . . . , 2 ] = 0 . n→∞ qn−1

(37)

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C. Elsner and C. R. Havens



For even n the bound for .H Q n (z) in (36) follows with Lemma 3.4 after multiplying (37) with .2/n, so that .2(pn−1 − qn−2 )/n ∈ Z. We recall the inequalities from Corollary 3.1. The inequality in the following theorem results from the right-hand inequality in (17) and from (36). Theorem 3.5 We have for increasing even numbers n,    z − Q n (z)  

.

n2 H 2

1

. Q n (z)

(38)

In terms of evaluating the approximation quality of .|z − Q n (z)| with the heights H (Q n (z)) the quadratic convergents .Q n (z) approximate .z better than the rational numbers .pn /qn for even n, which admit only an approximation quality of the form

.

 1 pn  1   z −   qn nqn2 nH 2 (pn /qn )

.

(cf. formula (1) in Sect. 2), where .H (pn /qn ) = pn > qn , because .Pn (X) = qn X − pn and .gcd(pn , qn ) = 1. Let n be an even number. With formula (9) we can justify the second of the following inequalities: n > n−1 

.

ln(qn−1 ) . ln ln(qn−1 )

e The function .ln(x)/ ln ln(x) is strictly increasing

for .x ≥ e = 15.154 . . . . Let .C > 0 be a constant in (36) such that .H Q n (z) < 2C zqn−1 /n. Now, choosing n

sufficiently large, we have .ee < H Q n (z) < 2C zqn−1 /n < qn−1 , and obtain



ln H (Q n (z)) ln(2C zqn−1 /n)

> .n  ln ln(2C zqn−1 /n) ln ln H (Q n (z)) for all large even n. This bound is now used to further enlarge the right-hand side in (38) by substituting for n. Corollary 3.6 For all large even integers n we have   . z − Q n (z)  



2

ln ln H (Q n (z))

. ln H (Q n (z)) H (Q n (z))

Diophantine Analysis Around .[1, 2, 3, . . . ]

131

3.4 Proof of Lemma 3.4 Lemma 3.7 Let n ≥ 2 be an even integer. Then we have

qn−1 ≡

.

pn−1 ≡ qn−2 ≡

.

pn−2 ≡

.

n/2 (mod n) , 0 (mod n) ,

if n ≡ 0 (mod 4) , if n ≡ 2 (mod 4) .

n/2 + 1 (mod n) , 1 (mod n) , 0 (mod n) , n/2 (mod n) ,

if n ≡ 4 (mod 8) , if n ≡ 0, 2, 6 (mod 8) .

if n ≡ 0 (mod 4) , if n ≡ 2 (mod 4) .

(39)

(40)

(41)

We prove this lemma in Sect. 3.5. But we can use Lemma 3.7 to complete the proof of Lemma 3.4: From (40) we conclude on pn−1 − qn−2 ≡ 0 (mod n), which 

together with (39) and (41) implies Lemma 3.4.

3.5 Proof of Lemma 3.7 We demonstrate the arguments for the congruences in (40), the remaining congruences in (39) and (41) can be proven similarly. Before starting to prove (40), we need explicit formulas for .pn and .qn , which have the form of sums over binomial coefficients. Lemma 3.8 We have for all .n ≥ 0, pn =

(n+1)/2 

.

 (n − 2k + 1)!

k=0

qn =

n/2  k=0



n−k (n − 2k)! k

 n−k+1 2 ,. k



 n−k+1 . k+1

Proof Lemma 3.8 states a special case for Corollary 7 in [4].

(42)

(43)



With the numbers .pi and .qi for .i = 0, 1, 2, 3 obtained from Lemma 3.8, we check that (40) holds for .n = 2 and .n = 4. From now on we write .n instead of .n/2 for even integers .n ≥ 6.

132

C. Elsner and C. R. Havens

Now we prove the two congruences in (40) for the numbers .pn−1 and .qn−2 . First we treat .pn−1 using (42):   n n   n−k 2 (n − 2k)! =: ck k 

pn−1 =

.



k=0

(44)

k=0

with   (n − k)! n − k .ck = k! k =

  (n − 3)! · (n − 2)(n − 1)n · (n + 1) . . . (n − k) n − k . k! k

(45)

Case 1.1 .0 ≤ k ≤ n − 3. Note that .n ≥ 6 implies .n − 3 ≥ 0. (i) By the condition of Case 1.1, .k! divides .(n − 3)!. (ii) Because of .(n − 2)(n − 1) ≡ 0 (mod 2) we have .(n − 2)(n − 1)n ≡ 0 (mod n). (iii) By the condition .k ≤ n − 3 we have in (45): .n − k ≥ n + 3. Altogether we obtain ck ≡ 0 (mod n) .

.

(46)

Case 1.2 .k = n − 2. Now .ck has the form  n + 2 2 (n + 4)2 (n + 2)2 n(n − 2)2 n. = 4!  = n −2 3 · 211 

cn −2

.

(47)

(i) 3 divides .(n + 4)(n + 2)n. (ii) Exactly one of the four numbers .n + 4, n + 2, n, n − 2 is divisible by .23 , another by .22 , the remaining two are each divisible by 2. In total, .3 · 211 divides 2 2 2 .(n + 4) (n + 2) n(n − 2) . So for (47) we get cn −2 ≡ 0 (mod n) .

.

(48)

Case 1.3 .k = n − 1.  cn −1 = 2!

.

n + 1 n − 1

2 =

(n + 2)2 n n. 25

(49)

Diophantine Analysis Around .[1, 2, 3, . . . ]

133

Case 1.3.1 .n ≡ 4 (mod 8). Then, .24 divides .(n + 2)2 n, but this does not hold for 5 .2 . Thus, (49) yields cn −1 ≡

.

n (mod n) . 2

(50)

Case 1.3.2 .n ≡ 0, 2, 6 (mod 8). Under this assumption, .25 divides .(n + 2)2 n, and so we get for (49) cn −1 ≡ 0 (mod n) .

(51)

  n = 1. n

(52)

.

Case 1.4 .k = n . cn = 0!

.

Finally, (40) for .pn−1 follows from (44), (46), (48), (50), (51), and (52). The following considerations will show that the same congruences hold for .qn−2 . From formula (43) we obtain qn−2 =

 −1 n

.

k=0



 −1   n n−k−2 n−k−1 (n − 2k − 2)! =: dk . k k+1

Case 2.1 .0 ≤ k ≤ n − 3. We have   (n − k − 2)! n − k − 1 .dk = k! k+1 =

(53)

k=0

(54)

  (n − 3)! · (n − 2)(n − 1)n · (n + 1) · · · (n − k − 2) n − k − 1 . k! k+1

Note that .n − k − 2 ≥ n + 1. As above in Case 1.1, the hypothesis of Case 2.1 guarantees dk ≡ 0 (mod n) .

.

Case 2.2 .k = n − 2. We obtain      n n +1 (n − 2)n(n + 2) n. .dn −2 = 2! = n − 2 n − 1 25

(55)

(56)

134

C. Elsner and C. R. Havens

Case 2.2.1 .n ≡ 4 (mod 8). Although .24 divides .(n − 2)n(n + 2), this does not hold for .25 . Thus, (56) yields dn −2 ≡

.

n (mod n) . 2

(57)

Case 2.2.2 .n ≡ 0, 2, 6 (mod 8). Now, .25 divides .(n − 2)n(n + 2). We obtain from (56), dn −2 ≡ 0 (mod n) .

.

(58)

Case 2.3 .k = n − 1. This results in  dn −1 = 0!

.

  n − 1 n = 1. n − 1 n

(59)

Finally, (40) for .qn−2 follows from (53), (55), (57), (58), and (59). This completes the proof of (40) in Lemma 3.7.



4 Error Sums of the Zopf-Number 4.1 Preliminaries Let .a ≥ 0 and .b ≥ 1 be integers. We define the numbers

zba := a + 1 +

.

b| b| b| + + + ... |a + 2 |a + 3 |a + 4

(60)

by their irregular .(b > 1) or regular .(b = 1) continued fraction expansion. In particular, .z1a = [a + 1, a + 2, a + 3, . . . ], and .z = z10 = [1, 2, 3, . . . ] gives the Zopf-number. The numbers ∞ 



E zba := (−1)m zba qm − pm .

.

(61)

m=0 ∞ 

b

E ∗ zba := za qm − pm .

(62)

m=0 ∞ 



(−1)m b za qm − pm . E f ac zba := m!

(63)

∞ 

1 b E ∗f ac zba := za qm − pm m!

(64)

m=0

m=0

Diophantine Analysis Around .[1, 2, 3, . . . ]

135

are called error sums of .zba , where .pm /qm are the convergents obtained from the regular or irregular continued fraction of .zba . Both the numerators .pm and denominators .qm satisfy the recurrence relation um+2 = (m + a + 3)um+1 + bum

.

(m ≥ 0)

(65)

with

.

p0 = a + 1 , q0 = 1 , p1 = a 2 + 3a + b + 2 , q1 = a + 2 , p2 = a 3 + 6a 2 + 2ab + 11a + 4b + 6 , q2 = a 2 + 5a + b + 6 ,

⎫ ⎬ ⎭

(66)

see [5, § 2, (12)]. Let us first consider two general types of error sums, namely the E and the .E ∗ function:

.

∞ 



E ∗ ξ := ξ qm − pm ,

.

∞   

ξ qm − pm  . E ξ :=

m=0

(67)

m=0

The sums extend in each case over all convergents .pm /qm of a real number .ξ . The theory of these two functions shows us that they have a fractal appearance, [3], [2]. The function .E ∗ was first studied in more detail by J.N. Ridley and G. Petruska in 2000, [7]. In 1992 Petruska had already used a special irrational number and its error sum .E ∗ to explicitly construct a so-called q-series with a given radius of convergence greater than 1, [6]. We list some properties of these error sums. 1. .E (ξ ) and .E ∗ (ξ ) are continuous functions at every irrational point .ξ , and discontinuous functions at every rational point .ξ . 2. The range of the function .E√is the set of all real numbers between zero and the Golden Number .G = (1 + 5)/2, whereas the range of the function .E ∗ is the set of all real numbers between 0 and 1. 3. Both functions are periodic with period 1. 4. The error sums satisfy simple functional equations, E ∗ (ξ ) + E ∗ (1 − ξ ) =

.

E (ξ ) − E (1 − ξ ) =

1 − ξ if 0 < ξ < 1/2 , ξ if 1/2 < ξ < 1 , ξ − 1 if 0 < ξ < 1/2 , ξ if 1/2 < ξ < 1 .

136

C. Elsner and C. R. Havens

5. From 1. and 2. it follows that .E (ξ ) and .E ∗ (ξ ) are Lebesgue integrable. We have 

1

.

E ∗ (α) dα =

3 , 8

1

π 2 ln 2 5 3ζ (2) ln 2 5 − = − = 0.79778798 . . . . 2ζ (3) 8 4ζ (3) 8

0



E (α) dα =

0

The functions f (x) :=

∞ 

.

(zba qm − pm ) · x m

(68)

xm m!

(69)

m=0

and g(x) :=

∞ 

.

(zba qm − pm ) ·

m=0

are called ordinary generating functions and exponential generating functions, respectively, of the error terms .zba qm − pm . From here we make use of Bessel functions, where .μ ≥ 0 is an integer. Jμ (x) : Yμ (x) : . Iμ (x) :

Bessel function of the first kind , Bessel function of the second kind , modified Bessel function of the first kind ,  x −μ Kμ (x) := (μ + 1) Jμ (x) . 2

For example, we have b .za

√ √ Ia (2 b) = b· √ . Ia+1 (2 b)

(70)

For real numbers t tending to infinity, we have the asymptotic behavior  Jμ (t) ∼

.

 μπ π 2 cos t − − . πt 2 4

(71)

The following relationships between Bessel functions will play an essential role in our investigations.

Diophantine Analysis Around .[1, 2, 3, . . . ]

137

Lemma 4.1 (i) Let .x ∈ C. The relationship between .Iμ (x) and .Jμ (x) is given by

−iπ μ/2 .Iμ (x) = e Jμ eiπ/2 x .

(72)

For real x, both .Iμ (x) and .Jμ (x), are either odd functions for odd integers μ ≥ 1, or even functions for even integers .μ ≥ 0. (ii) We have the recurrence relation .

.

2μ Iμ (x) = Iμ−1 (x) − Iμ+1 (x) x

(μ ≥ 1) .

(iii) Let .β, x ∈ C. Then we have  d  −μ . x Iμ (βx) = βx −μ Iμ+1 (βx) . dx

(73)

(74)

(iv) We have for .μ ≥ 1, Iμ (x) =

.

1 Iμ+1 (x) + Iμ−1 (x) . 2

(75) 

Proof See [1], [10].

4.2 Main Results For the error sums of .zba , the exponential generating function .g(x) from (69) turns out to be the theoretically more accessible object than the ordinary generating function .f (x) in (68), albeit with the restriction to .a = 0. However, .zb0 and .zba are related only by one explicitly given linear fractional transformation, namely

zba =

.

bzb0 qa−2 − bpa−2 pa−1 − zb0 qa−1

,

and therefore the restriction to .a = 0 is immaterial. So we start by listing results for g(x) for the error sums of .zb0 . Formula numbers marked with an asterisk refer to formulas obtained using MAPLE. For the sake of brevity, we also omit the proofs of these formulas that can be done with known standard methods of real analysis. All other unmarked formula numbers refer to statements that are either obvious or will be proven.

.

Theorem 4.2 Let .a = 0 and .b ≥ 1. The function .g(x) satisfies the differential equation (x − 1)g  + 3g  + bg = 0 .

.

(76)

138

C. Elsner and C. R. Havens

Moreover, we have g(x) =

.

√ √ I2 (2 b(1 − x)) b √ (1 − x)I1 (2 b)

(x ∈ R \ {1}) ,

√ b3 = lim g(x) = √ ,. x→1 2I1 (2 b) √ √ I2 (2 2b) b E f ac (z0 ) = g(−1) = b √ . 2I1 (2 b)

∗ b .E f ac (z0 )

(77)*

(78) (79)

Equation (78) follows from (77)* using the limit √ b I2 (2 b(1 − x)) = , . lim x→1 1−x 2 which can be obtained by replacing x in Iμ (x) =

.

∞  x μ 

2

k=0

 x 2k 1 k! (μ + k + 1) 2

√ by .2 b(1 − x). Corollary 4.3 For .b = 1 we have E ∗f ac (z) =

.

1 = 0.314339 . . . , . 2I1 (2)

√ I2 (2 2) = 0.583891 . . . . E f ac (z) = 2I1 (2)

(80)

(81)

The exponential generating function .g(x) on the right-hand side in (69) is a Taylor expansion around the point .x = 0. However, the expansion of this function around the point infinity is also particularly interesting. We present this expansion in the following theorem. Theorem 4.4 For positive increasing .x > 1 we have the asymptotic expansion √

√  sin 5π/4 + 2 b(x − 1) b · .g(x) = √ π I1 (2 b)x 5/4

 √

15 cos 5π/4 + 2 b(x − 1) + O x −9/4 . · + √ 7/4 16 I1 (2 b)x

(82)*

Diophantine Analysis Around .[1, 2, 3, . . . ]

139

For .b = 1 there are rational numbers .qm,1 and .qm,2 such that we have for all real numbers .x > 1 the series g(x) = √

.

∞    5π  √ 1 qm,1 sin +2 x−1 m+1/4 4 x π I1 (2) m=1

+

qm,2 x m+3/4

cos

 5π 4

 √ +2 x−1 .

(83)*

In particular, .q1,1 = 1 and .q1,2 = 15/16. We obtain a corollary from Theorems 4.2 and 4.4. Corollary 4.5 We have

.



lim g(x) = lim

x→∞ ∞

x→∞

g(x) dx = lim

x→∞

0

∞ 

(zb0 qm − pm )

m=0 ∞ 

xm = 0 ,. m!

(zb0 qm−1 − pm−1 )

m=1

xm = 1. m!

(84)

(85)

Moreover, for every .ε > 0 and sufficiently large .x > 1 we have √ 4  5/4  1 b .x g(x) < √ · √ +ε, π I1 (2 b)

(86)

where the numerical constant √ 4 1 b · .√ √ π I1 (2 b)

(87)

is best-possible. Now we turn to the function .f (x). Again, we allow arbitrary integers .a ≥ 0 and b ≥ 1 and we state our results for .f (x) in Theorems 4.6 and 4.7 without proofs. Also for these theorems only standard methods of real analysis are used.

.

Theorem 4.6 The function .f (x) satisfies the differential equation

x 2 f  + bx 2 + (a + 2)x − 1 f = bx − zba + a + 1 .

.

Moreover, we have f (x) =

.

 1 x a+2 ebx+1/x



x 1



bt − zba +a +1 t a ebt+1/t dt +eb+1 E ∗ (zba )



(x > 0)

140

C. Elsner and C. R. Havens

and f (x) =

.

  −1 

a bt+1/t 1 b a −b−1 b − bt− z +a+1 t e dt+(−1) e E ( z ) (x < 0). a a x a+2 ebx+1/x x

The function .e1/t is analytical for every real number t except .t = 0. Therefore, we have in Theorem 4.6 no common formula for all real numbers x. For .x = 0, the limit is given by  1 .f (0) = lim − ε→0+ ε a+2 ebε+1/ε =

zba q0

− p0 =

zba





bt − zba + a + 1 t a ebt+1/t dt

1 ε

− a − 1.

We end the listing of results with a theorem for an exponential sum, which becomes an error sum for minor convergents of .zb0 when .b = 1. Theorem 4.7 Let .a = 0, .b ≥ 1, and .k ∈ N. Moreover, let .Pk,m := kpm+1 + pm and .Qk,m := kqm+1 + qm , where .pm /qm are the convergents of .zb0 given by (65) and (66). Then we have h(k) :=

.

√ ∞  I  (2i bk) (k + 1)m b · (zb0 Qk,m − Pk,m ) = √ · 2 √ , m! k i I (2 b) 1 m=0  .

0

k

h(t) dt =

√

√ J2 (−2 bk) b· . √ I1 (2 b)

Example For .k = 1 we obtain from Theorem 4.7, √ ∞ 

bI2 (2i b) 2m b · z0 (qm+1 + qm ) − (pm+1 + pm ) = . √ . m! iI1 (2 b) m=0 √ √

b I1 (2i b) + I3 (2i b) . = √ 2iI1 (2 b) √

√ b J1 (2 b) − J3 (2 b) = . √ 2I1 (2 b) In the second last step, we first apply formula (75) with .μ = 2, and then (72) in order to obtain the final real expression.

Diophantine Analysis Around .[1, 2, 3, . . . ]

141

4.3 Proofs We prove the outstanding parts in this section (apart from Theorems 4.6 and 4.7), skipping the intermediate computations performed with MAPLE. In some places, however, we give individual intermediate steps in the procedure even for the MAPLE calculations. Proof of (76) and (77)* in Theorem 4.2 Let y(x) :=

.

∞  um m x , m!

(88)

m=0

where the integers .um are given recursively by (65) and (66) with .a = 0. Differentiating twice, we obtain the series y  (x) =

∞ 

.

m=1

um x m−1 (m − 1)!

and

y  (x) =

∞  m=2

um x m−2 . (m − 2)!

(89)

With an index shift on the right-hand side of (89) and application of the recursion formula (65), we obtain y  (x) =

.

∞ ∞ ∞ ∞    um+2 m  um um um m x = xm − 1 + 3 x m−1 + b x m! (m − 2)! (m − 1)! m! m=2

m=0 

m=1

m=0



= xy (x) + 3y (x) + by(x) . The last identity follows from (88) and (89), and (76) is proven. MAPLE calculates the general solution of the differential equation (76) as √ √



J2 2 b(x − 1) Y2 2 b(x − 1) .y(x) = C1 · + C2 · x−1 x−1

(90)*

with arbitrary constants .C1 and .C2 . We now calculate the special solutions .y(x) = g1 (x) and .y(x) = g2 (x), once for .um = pm and another time for .um = qm . In any case we obtain the two constants .C1 and .C2 from the initial conditions .g1 (0) = p0 = 1, .g1 (0) = p1 = b + 2, .g2 (0) = q0 = 1, and .g2 (0) = q1 = 2. All results are entered into (90)* and then merged to form the function .g(x) = zb0 g2 (x) − g1 (x), where .zb0 is expressed by (70) with .a = 0. After significant simplifications of the resulting terms, MAPLE calculates the final result given in (77)* on the right-hand side of the formula. 

142

C. Elsner and C. R. Havens

Proof of Corollary 4.5 There is nothing to prove for (84), (86), and (87), because the assertions follow directly from Theorem 4.4. In particular, the principal term in the asymptotic expansion of .x 5/4 g(x) in (82)* takes it’s maximum and minimum values for x satisfying  5π π + 2 b(x − 1) = + kπ (k = 1, 2, 3, . . . ) . 4 2 √ √ It remains to prove (85). Let .β := 2 b and .z := 1 − x. Then, by (77)*, .



∞

E :=

.

√

g(x) dx =

0

b I1 (β)

 0

∞

I2 (βz) dx . z2

(91)

Next, we apply formula (74) for .μ = 1 and so it takes the form I2 (βz) =

.

z d  I1 (βz)  . β dz z

Moreover, we have .

1 dz 1 = − √ = − , dx 2z 2 1−x

or

dx = −2z dz .

Overall, the integral in (91) can then be transformed as follows, √  i∞ √  ∞ 1 d  I1 (βz)  −2z d  I1 (βz)  b b dx = dz .E = I1 (β) 0 βz dz z I1 (β) 1 βz dz z = (72)

=

−1 I1 (β)



i∞

d 1

 I (βz)  −1  I1 (βz) z=i∞ 1 = z I1 (β) z z=1

 (71) −1

J1 (βt) −1  lim − I1 (β) = · − I1 (βt) = 1 . t→∞ I1 (β) t I1 (β)

This proves the identity in (85).



Diophantine Analysis Around .[1, 2, 3, . . . ]

143

5 Concluding Comments The results of this paper represent ideas born from the early correspondences of the authors. Further generalizations, to include a study of both .

∞  Ia/b (2/b) = a + kb k=0 and Ia/b+1 (2/b) √ √ Ia (2 b) b| b| b za = b + + ··· , √ =a+1+ |a + 2 |a +3 Ia+1 (2 b)

are forthcoming. Acknowledgments The authors thank the referee for numerous suggestions for improving details in the original manuscript.

References 1. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, 1965) 2. C. Elsner, A. Klauke, Errorsums for the values of the exponential function, Forschungsberichte der FHDW Hannover, Bericht Nr. 02014/01, 1–19; RS 8153 (2014, 1) 3. C. Elsner, M. Stein, On the value distribution of error sums for approximations with rational numbers. Integers 12, A66, 1–28 (2012) 4. J. Mc Laughlin, Some new families of Tasoevian and Hurwitzian continued fractions. Acta Arith. 135(3), 247–268 (2008) 5. O. Perron, Die Lehre von den Kettenbrüchen, Bd. 1 (Wissenschaftliche Buchgesellschaft, Darmstadt, 1977) 6. G. Petruska, On the radius of convergence of q-series. Indagationes Math. N.S. 3(3), 353–364 (1992) 7. J.N. Ridley, G. Petruska, The error-sum function of continued fractions. Indagationes Math. N.S. 11(2), 273–282 (2000) 8. C.L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preu. Akad. Wiss. Berlin, Kl. Math. Phys. Tech., 1, Gesammelte Abhandlungen, I, (1929), pp. 209–266 9. C.L. Siegel, Transzendente Zahlen (Bibliographisches Institut, Mannheim, 1967), Transcendental Numbers, Annals of Mathematic Studies, No. 16, Princeton University Press 10. G.N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, 1995)

On a Smoothed Average of the Number of Goldbach Representations Daniel A. Goldston and Ade Irma Suriajaya

Dedicated to the memory of Eduard Wirsing

Abstract Assuming the Generalized Riemann Hypothesis for the zeros of the Dirichlet L-functions with characters modulo q, we obtain a smoothed version of the average number of Goldbach representations for numbers which are multiples of a positive integer q. Such an average was first considered by Granville [11, 12] but without any smoothing factor. In this short article, we also show how the smoothing can be removed. Keywords Goldbach conjecture · L-function · Riemann zeta-function · Non-trivial zero

1 Introduction and Statement of Results In this paper all sums are over positive integers unless otherwise indicated. Let ψ2 (n) =



.

(m)(m ),

m+m =n

The second author was supported by JSPS KAKENHI Grant Numbers 18K13400 and 22K13895, and also by MEXT Initiative for Realizing Diversity in the Research Environment. D. A. Goldston Department of Mathematics and Statistics, San Jose State University, San Jose, CA, USA e-mail: [email protected] A. I. Suriajaya () Faculty of Mathematics, Kyushu University, Fukuoka, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Maier et al. (eds.), Number Theory in Memory of Eduard Wirsing, https://doi.org/10.1007/978-3-031-31617-3_10

145

146

D. A. Goldston and A. I. Suriajaya

where . is the von Mangoldt function, defined by .(n) = log p if .n = pm , p a prime and .m ≥ 1, and .(n) = 0 otherwise. Thus .ψ2 (n) counts the “Goldbach” representations of n as sums of both primes and prime powers, and these primes are weighted to make them have a “density” of 1 on the integers. The average number of Goldbach representations was first studied by Landau  [16] in 1900. Letting .π2 (n) = p+p =n 1 with p and .p primes, he proved using the prime number theorem with error term that    N2 N2 . . π2 (n) = +O 2 log2 N log3 N n≤N Landau’s result implies G(N) ∼

.

N2 , 2

where

G(N) :=



ψ2 (n).

(1)

n≤N

Hardy and Littlewood [14, 15] used this result in testing conjectures concerning ψ2 (n). More recently, Fujii [5–7] in 1991 proved the following theorem concerning the average number of Goldbach representations, which for the first time directly connected the error term in Landau’s formula to the zeros of the Riemann zetafunction and the Riemann Hypothesis (RH).

.

Theorem (Fujii) Assuming the Riemann Hypothesis, so that the complex zeros .ρ = β + iγ of the Riemann zeta-function .ζ (s) have .β = 1/2. Then G(N) =

.

 N ρ+1 4 4 N2 −2 + O(N 3 (log N ) 3 ). 2 ρ(ρ + 1) ρ

(2)

The error term above was improved by Bhowmik and Schlage-Puchta in [2] to O(N log5 N ) assuming RH, and this was refined by Languasco and Zaccagnini [17] who obtained the following result.

.

Theorem (Languasco-Zaccagnini) Assuming the Riemann Hypothesis, we have G(N) =

.

 N ρ+1 N2 −2 + O(N log3 N). 2 ρ(ρ + 1) ρ

(3)

As shown in [10], the above asymptotic can also be obtained by extending the method of [2]. Bhowmik and Schlage-Puchta in the previously mentioned paper [2] also proved unconditionally that G(N) =

.

 N ρ+1 N2 −2 + (N log log N), 2 ρ(ρ + 1) ρ

and therefore the error term in (3) is close to best possible.

On a Smoothed Average of the Number of Goldbach Representations

147

The Riemann Hypothesis was used in Fujii’s Theorem to handle the error term, and this obscures the dependence of the error term on the zeros. This has been addressed in the recent papers [1] and [3] which obtain an equivalence between the size of the error in (1) with a zero-free region for zeros of the Riemann zeta-function. Granville [11, 12] introduced the average number of Goldbach representations for integers which are multiples of q by defining1 

Gq (N ) :=

ψ2 (n),

.

for

2 ≤ q ≤ N.

(4)

n≤N q|n

When .q = 1 we write .G(N) = G1 (N ) for the case when we average over all Goldbach representations. Granville proposed a remarkable asymptotic formula for .Gq (N ), but at the time the error term in (2) appeared to interfere with the conjectured error of the proposed formula. This problem was removed by Bhowmik and Schlage-Puchta [2] and now the result (3) is available. Thus we have, assuming the Generalized Riemann Hypothesis (GRH) holds for all Dirichlet L-functions .L(s, χ ), with characters .χ (mod q), that Gq (N ) =

.

1 G(N) + O(N logC N), φ(q)

(5)

for some fixed constant C. This can be obtained by the method of [17] and [21], or the methods of [11] and [3]. The more general problem of considering averages of .ψ2 (n) in arithmetic progressions, or having each prime in the Goldbach representation of n in its own arithmetic progression has also been examined in [21] and [3]. Our goal in this note is to study a smoothed average for the number of Goldbach representations which are multiples of q. We define this average to be Fq (N ) =



.

ψ2 (n)e−n/N ,

(6)

n q|n

and when .q = 1 we let .F (N) := F1 (N ) for averaging over all positive integers. By smoothing with this power series weight, we are able to obtain for .Fq (N ) and .F (N) results corresponding to (5) and (3) for .Gq (N ) and .G(N) with minimal effort. Theorem 1.1 Suppose the Generalized Riemann Hypothesis holds for the Dirichlet L-functions .L(s, χ ) with characters .χ (mod q). Then for .q ≥ 2 Fq (N ) =

.

1 We

have .Gq (N ) = 0 if .q > N .

1 F (N) + O (N log N log q) , φ(q)

(7)

148

D. A. Goldston and A. I. Suriajaya

and when .q = 1 F (N) = N 2 − 2



.

(ρ)N ρ+1 + O(N),

(8)

ρ

where the sum runs over the non-trivial zeros of the Riemann zeta-function, and (s) is the gamma function.

.

The second equation actually depends on RH, which is included from our assumption on .L(s, χ0 ). The method we use is a combination of the methods of the papers [8] and [4]. We can use this method to prove (5) but the proof is much more complicated than the proof of Theorem 1.1. This paper is organized as follows. In Sect. 2 we set up the results needed in the proof of Theorem 1.1. These are actually proved in greater generality than necessary so that they also apply for .Gq (N ). In Sect. 3 we prove Theorem 1.1. Finally, in Sect. 4 we outline how to apply this method to .Gq (N ).

2 Lemmas Let Fq (z) :=



.

ψ2 (n)zn ,

(9)

n q|n

where we take, for .N ≥ 4, z = re(α),

.

r = e−1/N ,

and

e(α) = e2π iα .

(10)

We also have, on letting .F (z) = F1 (z), that F (z) = (z)2 ,

.

where

(z) :=

 n

To evaluate .Fq (z), we use (z, χ ) :=



.

n

where .χ is a Dirichlet character .(mod q).

χ (n)(n)zn ,

(n)zn .

(11)

On a Smoothed Average of the Number of Goldbach Representations

149

Lemma 2.1 For .N ≥ 4 and .q ≥ 2, we have Fq (z) =

.

1 φ(q)

  χ (−1)(z, χ )(z, χ ) + O (log N log q)2 .



(12)

χ (mod q)

If .q = 1 we have the identity (11). Proof (Proof of Lemma 2.1) For .N ≥ 4 and .q ≥ 2, we have Fq (z) =

 q|m+m



 ⎟⎜ ⎜  =⎝ + ⎠⎜ ⎝ m (m,q)=1

m m ≡−m (mod q)

m (m,q)>1

(m)(m )zm+m

m+m ≡0 (mod q)

⎞⎛

⎛ .





(m)(m )zm+m =



⎞

⎟ (m)(m )zm+m ⎟ ⎠

(13)

=: S1 + S2 . We introduce characters into .S1 using the relation, for .(a, q) = 1, .

1 φ(q)



χ (a)χ(n) = 1n≡a(mod q) .

χ (mod q)

Thus S1 =



.

m,m

⎛

⎝ 1 ϕ(q) 

1 = ϕ(q) =

1 ϕ(q)



⎞ χ (−m)χ(m )⎠ (m)(m )zm+m

χ (mod q)

 χ (mod q)



χ (−1)





χ (m)(m)z

m

 

m

χ (−1)(z, χ )(z, χ ).

χ (mod q)

For .S2 , we note that .(m , q) > 1, and therefore ⎞2  ⎟ ⎜ .S2  ⎝ (m)r m ⎠ . ⎛

m (m,q)>1

m



 



χ (m )(m )z

m

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D. A. Goldston and A. I. Suriajaya

We have 

(m)r m =

m (m,q)>1

 k (log p)r p p|q

k

⎛ ≤⎝

 p|q

.

⎞   2k log p⎠ e− N

⎛

≤ log q ⎝ ⎛

k



1+

2k ≤N



⎞ e

k − 2N

2k >N





log N ≤ log q ⎝ log 2

+



(14)

⎠ ⎞

e−j ⎠

j

 log N log q, and (12) follows. If .q = 1, we see in (13) that .S2 = 0 and F (z) =



.



(m)(m )zm+m = (z)2 .

m,m

 Let .χ0 denote the principal character, and define  E0 (χ ) :=

1,

if χ = χ0 ,

0,

otherwise.

.

Lemma 2.2 Let .z = e−w , so that from (10) we have .w = .q ≥ 1 (z, χ ) =



.

χ (n)(n)e−nw =

n

1 N

− 2π iα. Then for

E0 (χ )  − (ρ)w −ρ + O(log(2qN) log 2q), w ρ (15)

where the sum is over the non-trivial zeros of .L(s, χ ). In particular, if .w = for .χ = χ0 and .q ≥ 3 we have (r, χ ) =



.

n

χ (n)(n)e−n/N = −

 ρ

(ρ)N ρ + O(log(qN ) log q),

1 N

then

(16)

On a Smoothed Average of the Number of Goldbach Representations

151

and for .χ = χ0 and .q ≥ 2 

(z, χ0 ) =

(n)zn = (z) + O(log N log q).

.

(17)

n (n,q)=1

Finally, we have (r) =



.

(n)e−n/N = N −

n



 (ρ)N ρ − log 2π + O

ρ

1 N

 ,

(18)

with the sum over the non-trivial zeros of the Riemann zeta-function. The proof of (15) may be found in [18, Lemma 2]. It is stated there that GRH is assumed, but the proof holds unconditionally. See also [21, Lemma 2.1]. In this paper we only need the special cases where .z = r. We see (17) follows immediately from (14). In the explicit formula for .(r) (18), the error term is from [20, 12.1.1 Exercise 8(c)] which follows from the original proof in [13].

3 Proof of Theorem 1.1 We take .z = r = e−1/N in (9) and have  .Fq (r) = ψ2 (n)e−n/N = Fq (N ), n q|n

using the notation (6). By Lemma 2.1 we have for .q = 1 F (N) = (r)2 ,

.

(19)

and for .q ≥ 2, Fq (N ) =

.

1 φ(q)



  χ (−1)|(r, χ )|2 + O (log N log q)2 .

(20)

χ (mod q)

In Lemma 2.2 we use . (ρ)  e−|γ | and the estimate for all .t ∈ R  . 1  log(2q(|t| + 1)) t N 2 since then Fq (N ) 



.

n(log n)2 e−n/N  N 2 e−N



m2 e−m  1,

m

n>N 2 q|n

and the error term in (7) is larger than both main terms. Thus we assume henceforth that .2 ≤ q ≤ N 2 , so that, for example, we can say .1  log q  log N. Applying (20) and (22), we have ⎛ ⎞    1 1 (r, χ0 )2 + O ⎝ .Fq (N ) = N log2 q ⎠ + O (log N log q)2 φ(q) φ(q) χ =χ0

  1 (r, χ0 )2 + O N log2 q . = φ(q) Therefore, by (17), (19), and the bound .(r)  N , we have   1 ((r) + O(log N log q))2 + O N log2 q φ(q)   1 = (F (N) + O(N log N log q)) + O N log2 q) φ(q)

Fq (N ) = .

=

1 F (N) + O (N log N log q) . φ(q) 

Proof of (8) in Theorem 1.1 The idea here is that the smooth Fujii formula (8) is just the square of the explicit formula for the powerseries generating function for primes. By (18) in Lemma 2.2 we have .(r) = N − ρ (ρ)N ρ +O(1). Assuming  RH, we have from (21) that when .q = 1 that . ρ (ρ)N ρ  N 1/2 . Therefore by (19) on squaring we have  F (N) = (r) = N −

.

2

 ρ

2 (ρ)N + O(1) ρ

= N 2 −2



(ρ)N ρ+1 +O(N ).

ρ



On a Smoothed Average of the Number of Goldbach Representations

153

4 Transition from F q(N ) to Gq(N) In [8] we made the transition from .F (z) = (z)2 to .G(N) to obtain (3). A similar procedure works to make the transition from .Fq (z) to .Gq (N ) to obtain (5). Here we present only a sketch of the proof. The interested reader can fill in the details from [8]. The starting point is the formula, with .z = re(α) and .r = e−1/N , 

1

.

0

Fq (z)IN ( 1z ) dα



=

ψ2 (n)r

n



r

−n

1

e(α(n − n )) dα = Gq (N ),

0

n ≤N

n q|n



(23) where IN (z) :=



.

 zn = z

n≤N

1 − zN 1−z

 .

When .q = 1 we obtain  G(N) =

.

0

1

(z)2 IN ( 1z ) dα.

(24)

We now take .2 ≤ q ≤ N. Applying Lemma 2.1 to (23) yields Gq (N ) =

.

1 φ(q) +



1

0

(z, χ0 )2 IN ( 1z ) dα

 1   1  χ (−1) (z, χ )(z, χ )IN ( 1z ) dα + O log5 N φ(q) 0 χ =χ0

   1/2 1 2 1 G(N) + O max = |(z, χ )| min{N, α } dα χ =χ0 0 φ(q)   N log3 N + O(log5 N), +O φ(q)

(25)

where we have used (17) and (24), while the first and third error terms are obtained by noting that  .

0

1

 |IN ( 1z )| dα =

1/2

−1/2

 |IN ( 1z )| dα 

1/2 −1/2

1 min{N, |α| } dα  log N.

154

D. A. Goldston and A. I. Suriajaya

Following the proof of [8, Theorem 2], we can bound the integral .I in the first error term of (25) as 

I 

N 2k

.

klog N



2k N

|(z, χ )|2 dα,

(26)

0

where upon using Gallagher’s lemma [19, Lemma 1.9], we have 

1/2h

.

|(z, χ )|2 dα 

0

1 (I1 (N, h) + I2 (N, h)), h2

(27)

with 

   

h 

I1 (N, h) :=

.

0

χ (n)(n)e

2   dx, 

−n/N 

n≤x

and  I2 (N, h) :=

.

0

 2     −n/N   χ (n)(n)e   dx. x r + 1/2 (.1 ≤ j ≤ k).

Spherical Summations of Arithmetic Functions of the GCD and LCM

163

3 Main Results 3.1 Spherical Summations of Arbitrary Functions For functions .F, G : Nk → C (.k ≥ 1) consider their convolution .F ∗ G defined by 

(F ∗ G)(n1 , . . . , nk ) =

F (d1 , . . . , dk )G(n1 /d1 , . . . , nk /dk ),

.

(12)

d1 |n1 ,...,dk |nk

and the generalized Möbius function .μ(n1 , . . . , nk ) = μ(n1 ) · · · μ(nk ), which is the inverse of the k-variable constant 1 function under convolution (12). See the survey [20] on properties of (multiplicative) arithmetic functions of several variables. Our first result is the following. Theorem 3.1 Let .F : Nk → C be an arbitrary arithmetic function of k variables (.k ≥ 1) and assume that the multiple Dirichlet series ∞  .

n1 ,...,nk

(μ ∗ F )(n1 , . . . , nk ) nz11 · · · nzkk =1

is absolutely convergent provided that .zj ∈ C with .zj ≥ t (.1 ≤ j ≤ k), where 0 < t ≤ 1 is a real number.

.

(i) If .t = 1, then .

lim

x→∞



1 x k/2

F (n1 , . . . , nk ) =

n1 ,...,nk ∈N n21 +···+n2k ≤x

Vk BF,k , 2k

where .Vk is given by (6), and BF,k :=

∞ 

.

n1 ,...,nk =1

(μ ∗ F )(n1 , . . . , nk ) . n1 · · · nk

(ii) If .0 < t < 1, then  .

n1 ,...,nk ∈N n21 +···+n2k ≤x

F (n1 , . . . , nk ) =

Vk BF,k x k/2 + O(x (k−1+t)/2 ). 2k

For .k = 1 Part (i) of Theorem 3.1 is Wintner’s mean value theorem. See, e.g., [13, Th. 2.19], [16, p. 138]. Also, Part (i) is the analog of the corresponding result for summation of functions .F (n1 , . . . , nk ) with .n1 , . . . , nk ≤ x, obtained by Ushiroya

164

R. Heyman and L. Tóth

[22]. Note that if F is multiplicative, then BF,k =



.

p

1 1− p

k

∞  ν1 ,...,νk =0

F (pν1 , . . . , pνk ) . pν1 +···+νk

To give an application of Theorem 3.1 we remark that if .f : N → C is an arbitrary arithmetic function, then ∞

∞  .

n1 ,...,nk

f ((n1 , . . . , nk )) ζ (z1 ) · · · ζ (zk )  f (n) = , z ζ (z1 + · · · + zk ) nz1 +···+zk nz11 · · · nkk =1 n=1

see [20, Eq. (16)]. In particular, for .f (n) = τ (n), ∞  .

n1 ,...,nk

τ ((n1 , . . . , nk )) = ζ (z1 ) · · · ζ (zk )ζ (z1 + · · · + zk ), nz11 · · · nzkk =1

with .zj ∈ C, .zj > 1 (.1 ≤ j ≤ k). This shows that for the function .F (n1 , . . . , nk ) = τ ((n1 , . . . , nk )) we have ∞  .

n1 ,...,nk

(μ ∗ F )(n1 , . . . , nk ) = ζ (z1 + · · · + zk ), nz11 · · · nzkk =1

(13)

absolutely convergent for .z1 = · · · = zk = 1 (.k ≥ 2). We deduce that for .k ≥ 2, .

lim

1

x→∞ x k/2



τ ((n1 , . . . , nk )) =

n1 ,...,nk ∈N n21 +···+n2k ≤x

Vk ζ (k) 2k

and by (7), .

lim

x→∞

1 x k/2



τ ((n1 , . . . , nk )) = Vk ζ (k).

n1 ,...,nk ∈Z n21 +···+n2k ≤x

Moreover, series (13) is absolutely convergent if .zj > 1/k (.1 ≤ j ≤ k), hence by Part (ii) of Theorem 3.1 we deduce the formula with error term  .

n1 ,...,nk ∈N n21 +···+n2k ≤x

τ ((n1 , . . . , nk )) =

Vk ζ (k) k/2 x + O(x (k−1+1/k)/2+ε ), 2k

but this error can be improved, see Corollary 3.3.

(14)

Spherical Summations of Arithmetic Functions of the GCD and LCM

165

3.2 Estimates for Functions of the GCD We prove the following estimates. Theorem 3.2 Let .k ≥ 2 and let .f = g ∗ 1, where g is a bounded function. Then 

f ((n1 , . . . , nk )) = Vk D(g, k)x k/2 + Rk (x),

.

n1 ,...,nk ∈Z n21 +···+n2k ≤x

∞ g(n) √ x, .R3 (x)  x 517/1648+ε , .R4 (x)  where .D(g, k) = n=1 nk , .R2 (x)  x(log x)2/3 , and .Rk (x)  x k/2−1 if .k ≥ 5.  2 This applies, i.e., for the functions .f (n) = τ (n) and .f (n) = d|n μ (d), representing the number of squarefree divisors of n, giving better error terms than in (14). If .f (n) = τ (n), then we deduce the next results. Corollary 3.3 For every .k ≥ 2, 

τ ((n1 , . . . , nk )) = Vk ζ (k)x k/2 + Rk (x),

.

n1 ,...,nk ∈Z n21 +···+n2k ≤x

furthermore, for summation over natural numbers,  .

τ ((n1 , n2 )) =

n1 ,n2 ∈N n21 +n22 ≤x

√ 1√ π3 x− x log x + O( x), 24 2

and for .k ≥ 3,  .

τ ((n1 , . . . , nk )) =

n1 ,...,nk ∈N n21 +···+n2k ≤x

1  k/2 (k−1)/2 V +Rk (x), ζ (k)x − kV ζ (k − 1)x k k−1 2k

where .Vk is defined by (6), and .Rk (x) is given in Theorem 3.2. Theorem 3.4 Let .f = g ∗ id, where g is a bounded function. Then for .k = 2,  .

n1 ,n2 ∈Z n21 +n22 ≤x

f ((n1 , n2 )) =

3 D(g, 2)x log x + O(x), π

(15)

166

R. Heyman and L. Tóth

and for .k ≥ 3, 

f ((n1 , . . . , nk )) = Vk

.

n1 ,...,nk ∈Z n21 +···+n2k ≤x

where .D(g, k) = x k/2−1 if .k ≥ 5.

∞

g(n) n=1 nk ,

ζ (k − 1) D(g, k)x k/2 + Qk (x), ζ (k)

(16)

Q3 (x)  x, .Q4 (x)  x(log x)5/3 , and .Qk (x) 

.

Corollary 3.5 Let .f = g ∗ id, where g is a bounded function. Then 

f ((n1 , n2 )) =

.

n1 ,n2 ∈N n21 +n22 ≤x



f ((n1 , n2 , n3 )) =

.

n1 ,n2 ,n3 ∈N n21 +n22 +n23 ≤x

3 D(g, 2)x log x + O(x), 4π

9 π3 D(g, 3)x 3/2 − D(g, 2)x log x + O(x), 36ζ (3) 8π

and for .k ≥ 4, 

f ((n1 , . . . , nk ))

.

n1 ,...,nk ∈N n21 +···+n2k ≤x

1 . = 2k

  ζ (k − 1) ζ (k − 2) k/2 (k−1)/2 Vk + Qk (x), D(g, k)x − kVk−1 D(g, k − 1)x ζ (k) ζ (k − 1)

where .Qk (x) is given in Theorem 3.4. These results apply, e.g., for the functions .id = δ ∗ id, .σ = 1 ∗ id, .β = λ ∗ id (alternating sum-of-divisors function, cf. [19]), .ϕ = μ ∗ id (Euler function), .ψ = μ2 ∗ id (Dedekind function). Next we consider the function .fS,η implicitly defined by hS,η (n) := (μ ∗ fS,η )(n)

(log p)η , if n = pν a prime power with ν ∈ S, = 0, otherwise,

.

(17)

where .1 ∈ S ⊆ N and .η ≥ 0 is real. By Möbius inversion we obtain that for n = p pνp (n) ∈ N,

.

fS,η (n) =



.

d|n

hS,η (d) =



(log p)η #{ν : 1 ≤ ν ≤ νp (n), ν ∈ S},

p|n

where .fS,η (1) = 0 (empty sum). This function was introduced by the authors [9].

Spherical Summations of Arithmetic Functions of the GCD and LCM

167

If .S = N, then fN,η (n) :=



.

νp (n)(log p)η ,

p|n

which gives for .η = 1, .fN,1 (n) = log n, while .hN,1 (n) = (n) is the von Mangoldt function. For .η = 0 one has .fN,0 (n) = (n). Let .S = {1}. Then f{1},η (n) :=



.

(log p)η ,

p|n

and if .η = 0, then .f{1},0 (n)  = ω(n). If .η =1, then .f{1},1 (n) = log κ(n), where κ(n) = p|n p. Note that . n≤x h{1},1 (n) = p≤x log p = θ (x) is the Chebyshev theta function.

.

Theorem 3.6 If .1 ∈ S ⊆ N, .η ≥ 0 is real and .k ≥ 2, then 

fS,η ((n1 , . . . , nk )) = Vk

.

n1 ,...,nk ∈Z n21 +···+n2k ≤x

 (log p)η HS,k (p) x k/2 + Tk (x), p

 √ νk x(log x)η−1 for .k ∈ {2, 3}, .T4 (x)  where .HS,k := ν∈S 1/p , .Tk (x)  2/3 k/2−1 for .k ≥ 5. x(log x) for .k = 4, and .Tk (x)  x In particular, we deduce Corollary 3.7 Let .k ≥ 2, and let f be one of the functions .log n, .log κ(n), .ω(n), (n). Then

.

 .

f ((n1 , . . . , nk )) = Kf,k Vk x k/2 + Tk (x),

n1 ,...,nk ∈Z n21 +···+n2k ≤x

where Klog,k =

.

 log p , pk − 1 p

Kω,k =

.

 1 , pk p

and .Tk (x) is given in Theorem 3.6.

Klog κ,k =

 log p p

K ,k =

 p

pk

1 , pk − 1

,

168

R. Heyman and L. Tóth

3.3 Estimates for Functions of the LCM Given a fixed positive real number r let .A r denote the class of multiplicative arithmetic functions .f : N → C satisfying the following properties: there exist real constants .C1 , C2 such that .|f (p) − pr | ≤ C1 pr−1/2 for every prime p, and ν νr for every prime power .p ν with .ν ≥ 2. This class of functions was .|f (p )| ≤ C2 p defined by Hilberdink and Tóth [11]. Observe that the functions .id, σ, ϕ, ψ belong to the class .A 1 . Also, any bounded multiplicative function f such that .f (p) = 1 for every prime p, in particular 2 .f (n) = μ (n) is in the class .A 0 . We prove the following results. Theorem 3.8 Let .k ≥ 2 be a fixed integer and let .f ∈ A 1 be a function. Then 

f ([n1 , . . . , nk ]) =

.

n1 ,...,nk ∈N n21 +···+n2k ≤x

 Cf,k k k−1/4+ε x , + O x 2k k!

where Cf,k

.



 1 k = 1− p p

∞  ν1 ,...,νk =0

f (pmax(ν1 ,...,νk ) ) . p2(ν1 +···+νk )

Corollary 3.9 Let .k ≥ 2. Then  .

[n1 , . . . , nk ] =

Ck k x + Rk (x), 2k k!

[n1 , . . . , nk ] =

Ck k x + Rk (x), k!

n1 ,...,nk ∈N n21 +···+n2k ≤x

 .

n1 ,...,nk ∈Z n21 +···+n2k ≤x

where .Rk (x)  x k−1/4+ε for .k ≥ 3, .R2 (x)  x 3/2 log x, and Ck =

.



 1 k 1− p p

∞  ν1 ,...,νk =0

1 p2(ν1 +···+νk )−max(ν1 ,...,νk )

in particular, .C2 = ζ (3)/ζ (2), .C3 = ζ (3)ζ (5)

  p 1−

3 p2

+

4 p3

−

,

3 p4

+

1 p6

.

Spherical Summations of Arithmetic Functions of the GCD and LCM

169

Theorem 3.10 Let .k ≥ 2 be a fixed integer and let .f ∈ A 0 be a function. Then 

f ([n1 , . . . , nk ]) =

.

n1 ,...,nk ∈N n21 +···+n2k ≤x

 Vk Ef,k k/2 k/2−1/4+ε , x + O x 2k

where .Vk is given by (6) and Ef,k =

.



 1 k 1− p p

∞  ν1 ,...,νk =0

f (pmax(ν1 ,...,νk ) ) . pν1 +···+νk

Corollary 3.11 Let .k ≥ 2. Then  .

μ2 ([n1 , . . . , nk ]) =

n1 ,...,nk ∈N n21 +···+n2k ≤x

 Vk k/2 k/2−1/4+ε . x + O x (2ζ (2))k

4 Proofs 4.1 Proofs of the Lemmas Proof of Lemma 2.1 We have by the convolutional identity f = 1 ∗ (μ ∗ f ),  .



f ((n1 , . . . , nk )) =

n1 ,...,nk ∈Z n21 +···+n2k =n



(μ ∗ f )(d)

n1 ,...,nk ∈Z d|(n1 ,...,nk ) n21 +···+n2k =n



=

(μ ∗ f )(d)

a1 ,...,ak ∈Z d 2 (a12 +···+ak2 )=n

=



(μ ∗ f )(d)

d 2 |n

=





1

a1 ,...,ak ∈Z a12 +···+ak2 =n/d 2

(μ ∗ f )(d)rk (n/d 2 ).

d 2 |n

 

170

R. Heyman and L. Tóth

Proof of Lemma 2.3 By induction on k. Estimate (10) is true for k = 1: 

1=

.

n1 ∈N a1 n21 ≤x



 √ 1≤ n1 ≤ x/a1

1=

x + O(1). a1

Assume that (10) is true for k − 1, where k ≥ 2. Then 

1=

.

n1 ,...,nk ∈N a1 n21 +···+ak n2k ≤x

=





√ 1≤nk ≤ x/ak



 √ 1≤nk ≤ x/ak



+O

1

n1 ,...,nk−1 ∈N a1 n21 +···+ak−1 n2k−1 ≤x−ak n2k

Vk−1 (x − ak n2k )(k−1)/2 √ 2k−1 a1 · · · ak−1

 √ √ (x − ak n2k )(k−2)/2 ( a1 + · · · + ak−1 ) √ a1 · · · ak−1 (k−1)/2

Vk−1 a = k−1 √ k a1 · · · ak−1 2 ⎛

 √ 1≤nk ≤ x/ak

(18)

(x/ak − n2k )(k−1)/2

√ √ x (k−2)/2 ( a1 + · · · + ak−1 ) ⎝ +O √ a1 · · · ak−1

 √ 1≤nk ≤ x/ak

⎞ 1⎠ .

Here the error term is √ √ √ √ x (k−2)/2 ( a1 + · · · + ak−1 )  x 1/2 x (k−1)/2 ( a1 + · · · + ak−1 ) . .  = √ √ ak a1 · · · ak−1 ak a1 · · · ak−1 To estimate the main term we use Euler’s summation formula from Lemma 2.2. By choosing the function ψ(t) = (x 2 − t 2 )(k−1)/2 we deduce for k ≥ 2 that  .

 (x − n ) 2

2 (k−1)/2

x

=

(x 2 − t 2 )(k−1)/2 dt + O(x k−1 ),

0

1≤n≤x

where 

x

.

0

(x 2 − t 2 )(k−1)/2 dt = x k Ik

Spherical Summations of Arithmetic Functions of the GCD and LCM

171

with 

π/2

Ik :=

.

(cos t)k dt =

0

(2m−1)!! π (2m)!! · 2 , (2m)!! (2m+1)!! ,

if k = 2m, if k = 2m + 1.

Hence the main term in (18) is

.

(k−1)/2  Vk−1 ak k/2 (k−1)/2 I (x/a ) + O((x/a ) ) √ k k k 2k−1 a1 · · · ak−1

Vk−1 Ik x k/2 +O . = √ k−1 a1 · · · ak−1 ak 2



x (k−1)/2 √ a1 · · · ak−1

.

Here Vk−1 Ik = Vk /2, and the proof is complete.

 

Proof of Lemma 2.4 By induction on k, similar to the proof of Lemma 2.3, but here we only need the familiar formula  .

nk =

1≤n≤x

x k+1 + O(x k ) k+1

(k ∈ N).

Estimate (11) is true for k = 1:  .

n1 =

n1 ∈N a1 n21 ≤x

 √ 1≤ n1 ≤ x/a1

x n1 = +O 2a1



x a1

 .

Assume that (11) is true for k − 1, where k ≥ 2. Then  .

n1 · · · nk =

n1 ,...,nk ∈N a1 n21 +···+ak n2k ≤x

=

 √ 1≤nk ≤ x/ak

 √ 1≤nk ≤ x/ak



+O



nk

n1 · · · nk−1

n1 ,...,nk−1 ∈N a1 n21 +···+ak−1 n2k−1 ≤x−ak n2k

 nk

(x − ak n2k )k−1 − 1)!a1 · · · ak−1

2k−1 (k

 √ √ (x − ak n2k )k−3/2 ( a1 + · · · + ak−1 ) (19) a1 · · · ak−1

172

R. Heyman and L. Tóth

=



1 2k−1 (k − 1)!a1 · · · ak−1 ⎛ +O ⎝

√ 1≤nk ≤ x/ak

√ x k−3/2 ( a1

√ + · · · + ak−1 ) a1 · · · ak−1

nk (x − ak n2k )k−1  √ 1≤nk ≤ x/ak

⎞ nk ⎠ .

Here the error term is √ √ √ √ x k−1/2 ( a1 + · · · + ak−1 ) x k−3/2 ( a1 + · · · + ak−1 ) x , = .  a1 · · · ak−1 ak a1 · · · ak−1 ak and for the sum S in the main term we have  nk (x − ak n2k )k−1 .S := √ 1≤nk ≤ x/ak

=

 √ 1≤nk ≤ x/ak

nk

  k−1  k − 1 k−1−j j 2j x (−1)j ak nk j j =0

  k−1  j j k−1 = (−1) x k−1−j ak j j =0

=



2j +1

√ 1≤nk ≤ x/ak

nk

    k−1  k − 1 k−1−j j (x/ak )j +1 x + O((x/ak )j +1/2 ) (−1)j ak j 2j + 2 j =0

=

 √ xk Ak + O x k−1/2 / ak , 2ak

where Ak :=

.

  k−1  (−1)j k − 1 j =0

j +1

j

=

1 . k

This gives that the term in (19) is xk . +O 2k k!a1 · · · ak showing that (11) is true for k.



√ x k−1/2 ak , a1 · · · ak  

Spherical Summations of Arithmetic Functions of the GCD and LCM

173

4.2 Proof of the Generalization of Wintner’s Theorem Proof of Theorem 3.1 We have by the identity F = 1∗ (μ∗ F ), here with functions of k variables, where μ(n1 , . . . , nk ) = μ(n1 ) · · · μ(nk ), 

SF,k (x) :=

F (n1 , . . . , nk )

.

n1 ,...,nk ∈N n21 +···+n2k ≤x





=

(μ ∗ F )(d1 , . . . , dk )

n1 ,...,nk ∈N d1 |n1 ,...,dk |nk n21 +···+n2k ≤x



=

(μ ∗ F )(d1 , . . . , dk )

d1 ,a1 ...,dk ,ak ∈N d12 a12 +···+dk2 ak2 ≤x



=

(μ ∗ F )(d1 , . . . , dk )



1.

a1 ,...,ak ∈N d12 a12 +···+dk2 ak2 ≤x

d1 ,...,dk ∈N √ d1 ,...,dk ≤ x

By using Lemma 2.3 we deduce SF,k (x) =

.

 √ d1 ,...,dk ≤ x

(μ ∗ F )(d1 , . . . , dk ) 

Vk x k/2 +O × k 2 d1 · · · dk =

Vk x k/2 2k

 √ d1 ,...,dk ≤ x



x (k−1)/2 (d1 + · · · + dk ) d1 · · · dk



(μ ∗ F )(d1 , . . . , dk ) + RF,k (x), d1 · · · dk

where RF,k (x)  x (k−1)/2

.

 √ d1 ,...,dk ≤ x

|(μ ∗ F )(d1 , . . . , dk )|(d1 + · · · + dk ) , d1 · · · dk

and

.

k RF,k (x) 1   √ x k/2 x



√ j =1 d1 ,...,dk ≤ x

|(μ ∗ F )(d1 , . . . , dk )|dj . d1 · · · dk

(20)

174

R. Heyman and L. Tóth

(i) Assume that the series ∞  .

d1 ,...,dk =1

|(μ ∗ F )(d1 , . . . , dk )| d1 · · · dk

is convergent. Then for a small ε > 0 split the inner sum in (20) for j = 1 (and similarly for every j ) in two parts: 

1 √ x

.

√ d1 ,...,dk ≤ x



1 = √ x

√ d1 ≤ε x√ d2 ,...,dk ≤ x

1 +√ x

√

√ ε x x

d1 ,...,d √k d1 > x

∞ 

≤ x (t−1)/2

d1 ,...,dk =1

x

(t−1)/2

|(μ ∗ F )(d1 , . . . , dk )| d1t d2 · · · dk

,

since the latter series converges. This gives the error O(x (k−1+t)/2 ). For the error term in (21) by taking d1 in the numerator (similarly for d2 , . . . , dk ),  .

√ d1 ,...,dk ≤ x

|(μ ∗ F )(d1 , . . . , dk )|d1 = d1 · · · dk

 √ d1 ,...,dk ≤ x

|(μ ∗ F )(d1 , . . . , dk )|d1t d1t d2 · · · dk

∞ 

≤ x t/2

d1 ,...,dk =1

|(μ ∗ F )(d1 , . . . , dk )| d1t d2 · · · dk

 x t/2 , the latter series (the same as above) being convergent. This gives the same error, namely O(x (k−1+t)/2 ), and completes the proof.  

4.3 Proofs of the Results for Functions of the GCD According to (9) and (5), for every arithmetic function f , with .f (0) = 0 one has Sf,k (x) :=



.

n1 ,...,nk ∈Z n21 +···+n2k ≤x

f ((n1 , . . . , nk ))

176

R. Heyman and L. Tóth



=

(μ ∗ f )(d)rk (e)

d 2 e≤x

=



√ d≤ x

(μ ∗ f )(d)

(22)



rk (e)

e≤x/d 2

  (μ ∗ f )(d) + (μ ∗ f )(d)Pk (x/d 2 ). k d √ √

= Vk x k/2

d≤ x

d≤ x

 Proof of Theorem 3.2 Assume that .f = g ∗ 1, that is, .f (n) = d|n g(d) (.n ∈ N), where g is a bounded function with .|g(n)| ≤ K (.n ∈ N). Then .μ∗f = μ∗g∗1 = g. Hence .|(μ ∗ f )(n)| ≤ K for every .n ∈ N, and the series .

∞  (μ ∗ f )(n)

=

nk

n=1

∞  g(n)

nk

n=1

is absolutely convergent for every .k ≥ 2. From (22) we obtain Sf,k (x) = Vk x k/2

∞  g(d)

.

d=1

dk

⎛ + O ⎝x k/2

⎞ ⎞ ⎛   1 ⎠+O⎝ Pk (x/d 2 )⎠ . √ dk √

d> x

d≤ x

√ Here the first error term is .O( x) and using the known estimates for .Pk (x) given in Sect. 2.1 we obtain the indicated error terms by usual elementary estimates.   Proof of Corollary 3.3 For the second  part, namely for summation over the natural numbers, use (8) and the estimate . n≤x τ (n) = x log x + O(x) (this is sufficient).   Proof of Theorem 3.4 Now let .f = g ∗ id, where g is a bounded function with |g(n)| ≤ K (.n ∈ N). Then   .μ ∗ f = μ ∗ g ∗ id = g ∗ ϕ and .|(μ ∗ f )(n)| ≤ ϕ(d)|g(n/d)| ≤ K d|n d|n ϕ(d) = Kn for every .n ∈ N. This shows that for .k ≥ 3 the series .

.

∞  (μ ∗ f )(n) n=1

nk

=

∞  (g ∗ ϕ)(n) n=1

nk

∞

=

ζ (k − 1)  g(n) ζ (k) nk n=1

is absolutely convergent, and from (22) we obtain ⎛ ⎞ ⎞ ⎛ ∞    1 ζ (k − 1) g(n) k/2 ⎠+O ⎝ .Sf,k (x) = Vk x +O ⎝x k/2 dPk (x/d 2 )⎠ . ζ (k) nk √ d k−1 √ n=1

d> x

d≤ x

Spherical Summations of Arithmetic Functions of the GCD and LCM

177

Here the first error term is .O(x) and use the known estimates for .Pk (x) given in Sect. 2.1. This proves (16). If .k = 2, then ⎞ ⎛  (g ∗ ϕ)(d)  2 dP2 (x/d )⎠ . .Sf,2 (x) = V2 x +O⎝ d2 √ √ d≤ x

d≤ x

Using the known estimate  ϕ(n) .

n≤x

n2

=

6 log x + O(1) π2

(sufficient here in this form) we deduce

.

 (g ∗ ϕ)(n) n≤x

n2

 g(d)  ϕ(e) 6 = = 2 2 2 d e π d≤x

e≤x/d



∞  g(d) d=1

log x + O(1),

d2

which leads, together with .V2 = π and .P2 (x)  x 517/1648+ε to formula (15).

 

Proof of Corollary 3.5 Follows by (8) and the estimate

.

∞  x 2  g(n) + O(x log x), (g ∗ id)(n) = 2 n2 n≤x n=1

 

valid for every bounded function g. Proof of Theorem 3.6 For the function .fS,η we have by (17), .

 1  (μ ∗ fS,η )(d)  (log p)η  (log p)η = = k kν d p pkν √ √ ν √ p ≤ x ν∈S

d≤ x

=

 √ p≤ x

1≤ν≤m ν∈S

p≤ x

⎛

⎞

 ⎜ (log p)η ⎜ ⎝HS,k (p) −

ν≥m+1 ν∈S

1 ⎟ ⎟, pkν ⎠

(23)

x where .m =: 2log log p , and for every prime p, ∞

.

 1  1 1 1 , ≤ HS,k (p) := ≤ = k k kν kν p p p p −1 ν∈S

ν=1

(24)

178

R. Heyman and L. Tóth

using that .1 ∈ S. Here  .

√ p≤ x

(log p)η HS,k (p) =



(log p)η HS,k (p) −

p

 √ p> x

(log p)η HS,k (p),

(25)

where the series is absolutely convergent by (24), and the last sum is .

 (log p)η  (log p)η (log x)η−1  ,  pk x (k−1)/2 √ √ pk − 1



p> x

p> x

see [9, Lemma 3.3]. Also, A1 :=

.

=

 √ p≤ x



(log p)η

ν≥m+1 ν∈S



≤

1 x k/2

p≤ x

ν≥m+1

(log p)η . − 1)

(26)

√ p km (p k p≤ x

By the definition of m we have .m > sum in (26) is .

  1 1 (log p)η  kν p pkν √

log x 2 log p

− 1, hence .pkm >

x k/2 . pk

Thus the last

 pk (log p)η √ 1  1  k/2 (log p)η ≤ k/2 (log x)η π( x), k p −1 x x √ √

p≤ x

p≤ x

hence A1 

.

(log x)η−1 , x (k−1)/2

(27)

√ √ using .η ≥ 0 and the estimate .π( x)  logxx . We deduce by (22), (23), (25), (26) and (27) that

SfS,η ,k (x) = Vk x k/2

.

+



 √ (log p)η HS,k (p) + O( x(log x)η−1 ) p

√ pν ≤ x

(log p)η Pk (x/p2ν ).

(28)

ν∈S

Here the last sum can be estimated by using the known estimates for .Pk (x) given in Sect. 2.1. For example, if .k = 2, then .P2 (x)  x ϑ with .ϑ := 517/1648 + ε. We

Spherical Summations of Arithmetic Functions of the GCD and LCM

179

x deduce that the last sum in (28) is, with the notation .m =: 2log log p of above,

.



 √ pν ≤ x

(log p)η

 x ϑ   (log p)η  1 η ϑ ϑ = x ≤ x (log p) p2ν p2νϑ p2ϑ √ √ ν≤m p≤ x

 xϑ

p≤ x

√ (log x)η−1 = x(log x)η−1 , x (ϑ−1)/2

by using [9, Lemma 3.4]. The cases .k = 3, .k = 4 and .k ≥ 5 are similar, and lead to the stated error terms.  

4.4 Proofs of the Results for Functions of the LCM Proof of Theorem 3.8 Let f be a function in class A 1 . From Lemma 2.5 with r = 1 we deduce the convolutional identity  .f ([n1 , . . . , nk ]) = j1 · · · jk hf,k (d1 , . . . , dk ). j1 d1 =n1 ,...,jk dk =nk

Therefore



S :=

.

f ([n1 , . . . , nk ])

n1 ,...,nk ∈N n21 +···+n2k ≤x



=

j1 · · · jk hf,k (d1 , . . . , dk )

j1 ,d1 ,...,jk ,dk ∈N j12 d12 +···+jk2 dk2 ≤x

=



√ 1≤d1 ,...,dk ≤ x



hf,k (d1 , . . . , dk )

j1 · · · jk .

j1 ,...,jk ∈N d12 j12 +···+dk2 jk2 ≤x

By Lemma 2.4 we have 

S=

.



xk +O hf,k (d1 , . . . , dk ) 2k k!d12 · · · dk2 √

1≤d1 ,...,dk ≤ x

xk = k 2 k!

⎛

∞ 

hf,k (d1 , . . . , dk )

d1 ,...,dk =1

d12 · · · dk2

⎛

+O ⎝x k−1/2

+ ⎝x k



x k−1/2 (d1 + · · · + dk ) d12 · · · dk2

 |hf,k (d1 , . . . , dk )|

d1 ,...,dk

d12 · · · dk2



|hf,k (d1 , . . . , dk )|(d1 + · · · + dk )

√ 1≤d1 ,...,dk ≤ x

d12 · · · dk2



⎞ ⎠

⎞ ⎠

(29)

180

R. Heyman and L. Tóth

 √ where means that d1 , . . . , dk ≤ √ x does not hold. That is, there exists at least one m (1 ≤ m ≤ k) such that dm > x. Without loss of generality, we can suppose that m = 1. We obtain for 0 < ε < 1/4 that  |hf,k (d1 , . . . , dk )|

.

d1 ,...,d √k d1 > x

d12 · · · dk2

 |hf,k (d1 , . . . , dk )|  d1 1/2−2ε ≤ √ x d12 · · · dk2 d ,...,d 1 √k d1 > x

≤x

∞ 

ε−1/4

|hf,k (d1 , . . . , dk )|

d1 ,...,dk =1

3/2+2ε 2 d2 · · · dk2

d1

 x ε−1/4 ,

since the latter series converges by Lemma 2.5. For the error term in (29) by taking d1 in the numerator (similarly for d2 , . . . , dk ),

.



|hf,k (d1 , . . . , dk )|d1

√ 1≤d1 ,...,dk ≤ x

d12 · · · dk2

=



1/2+2ε

|hf,k (d1 , . . . , dk )|d1

√ 1≤d1 ,...,dk ≤ x ∞ 

≤ x 1/4+ε

d1 ,...,dk =1

3/2+2ε 2 d2 · · · dk2

d1

|hf,k (d1 , . . . , dk )| 3/2+2ε 2 d2 · · · dk2

d1

 x 1/4+ε , the latter series, the same as above, being convergent. This completes the proof.

 

Proof of Corollary 3.9 If k = 2, then Theorem 3.8 provides the error O(x 7/4+ε ). We show that for k = 2 and the function f (n) = n, the error term is O(x 3/2 log x). To this end we remark that D(z1 .z2 ) :=

.

∞  [n1 , n2 ] ζ (z1 + z2 − 1) . z1 z2 = ζ (z1 − 1)ζ (z2 − 1) ζ (z1 + z2 − 2) n n n ,n =1 1 2 1

(30)

2

To present a short direct proof of this identity, write n1 = da1 , n2 = da2 with (a1 , a2 ) = 1. Then [n1 , n2 ] = da1 a2 and we deduce D(z1 , z2 ) =

∞ 

.

d,a1 ,a2 =1 (a1 ,a2 )=1

da1 a2 = (da1 )z1 (da2 )z2

∞  d,a1 ,a2 =1

da1 a2 (da1 )z1 (da2 )z2

 δ|(a1 ,a2 )

μ(δ)

Spherical Summations of Arithmetic Functions of the GCD and LCM

181

and by denoting a1 = δb1 , a2 = δb2 , ∞ 

D(z1 , z2 ) =

.

d,δ,b1 ,b2 =1

=

∞  d=1

dδ 2 b1 b2 μ(δ) (dδb1 )z1 (dδb2 )z2 ∞ 

1 d z1 +z2 −1

δ=1

∞ 

μ(δ) δ z1 +z2 −2

1

∞ 

z1 −1 b1 =1 b1 b2 =1

1 b2z2 −1

,

giving (30).   Now consider the functions h(n) = d|n dμ(d) = p|n (1 − p), and h(n1 , n2 ) =

nh(n),

.

0,

if n1 = n2 = n, otherwise,

(31)

satisfying

.

∞  h(n1 , n2 ) ζ (z1 + z2 − 1) . z1 z2 = ζ (z1 + z2 − 2) n n 1 2 n ,n =1 1

2

This shows that in the above proof of Theorem 3.8, in the case k = 2 and f (n) = n one has hf,2 (n1 , n2 ) = h(n1 , n2 ), defined by (31). Now, following that proof,  .

√ d1 ,d2 ≤ x

 |h(d)|  1 |h(d1 , d2 )|(d1 + d2 ) =2   log x, 2 2 2 d d1 d2 √ √ d d≤ x

d≤ x

and  |h(d)|  1  |h(d1 , d2 )| 1 =  √ , 2 2 3 2 d x d1 d2 √ √ d d ,d

.

1 √2 d1 > x

d> x

d> x

 

leading to the stated error term.

Proof of Theorem 3.10 Similar to the proof of Theorem 3.8. Let f be a function in class A 0 . From Lemma 2.5 with r = 0 we deduce the identity f ([n1 , . . . , nk ]) =



.

j1 d1 =n1 ,...,jk dk =nk

hf,k (d1 , . . . , dk ).

182

Hence  .

R. Heyman and L. Tóth

f ([n1 , . . . , nk ]) =

n1 ,...,nk ∈N n21 +···+n2k ≤x

hf,k (d1 , . . . , dk )

j1 ,d1 ,...,jk ,dk ∈N j12 d12 +···+jk2 dk2 ≤x

=

and use Lemma 2.3.





√ 1≤d1 ,...,dk ≤ x

hf,k (d1 , . . . , dk )



1,

j1 ,...,jk ∈N d12 j12 +···+dk2 jk2 ≤x

 

References 1. T.M. Apostol, Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics (Springer, New York, 1976) 2. B.C. Berndt, S. Kim, A. Zaharescu, The circle problem of Gauss and the divisor problem of Dirichlet - still unsolved. Amer. Math. Mon. 125, 99–114 (2018) 3. O. Bordellès, L. Tóth, Additive arithmetic functions meet the inclusion-exclusion principle. Lith. Math. J. 62, 150–169 (2022) 4. J. Bourgain, N. Watt, Mean square of zeta function, circle problem and divisor problem revisited. Preprint (2017), 23 pp. https://arxiv.org/abs/1709.04340 5. F. Chamizo, E. Cristóbal, A. Ubis, Visible lattice points in the sphere. J. Number Theory 126, 200–211 (2007) 6. D. Essouabri, C. Salinas Zavala, L. Tóth, Mean values of multivariable multiplicative functions and applications to the average number of cyclic subgroups and multivariable averages associated with the LCM function. J. Number Theory 236, 404–442 (2022) 7. R. de la Bretèche, Estimation de sommes multiples de fonctions arithmétiques. Compos. Math. 128, 261–298 (2001) 8. E. Grosswald, Representations of Integers as Sums of Squares (Springer, New York, 1985) 9. R. Heyman, L. Tóth, On certain sums of arithmetic functions involving the GCD and LCM of two positive integers. Results Math. 76(1), Paper No. 49, 22 pp. (2021) 10. R. Heyman, L. Tóth, Hyperbolic summation for functions of the GCD and LCM of several integers. Ramanujan J. (2022). https://doi.org/10.1007/s11139-022-00681-2 11. T. Hilberdink, L. Tóth, On the average value of the least common multiple of k positive integers. J. Number Theory 169, 327–341 (2016) 12. T. Hilberdink, F. Luca, L. Tóth, On certain sums concerning the GCD’s and LCM’s of k positive integers. Int. J. Number Theory 16, 77–90 (2020) 13. A.J. Hildebrand, Introduction to Analytic Number Theory, Lecture Notes (2013). http://www. math.uiuc.edu/~hildebr/ant 14. A. Ivi´c, E. Krätzel, M. Kühleitner, W.G. Nowak, Lattice points in large regions and related arithmetic functions: recent developments in a very classic topic, in Elementare und analytische Zahlentheorie (Franz Steiner Verlag, Stuttgart, 2006), pp. 89–128 15. E. Krätzel, Lattice Points, Mathematics and its Applications, East European Ser. 33 (Kluwer, Dordrecht, 1988) 16. A.G. Postnikov, Introduction to Analytic Number Theory (American Mathematical Society, Providence, 1988) 17. W. Sierpi´nski, Elementary Theory of Numbers, 2nd edn. (North-Holland Publishing/Polish Scientific Publishers, Amsterdam/Warsaw, 1988)

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18. Y. Sui, D. Liu, Error term concerning number of subgroups of group Zm ×Zn with m2 +n2 ≤ x. Front. Math. China 17, 987–999 (2022) 19. L. Tóth, A survey of the alternating sum-of-divisors function. Acta Univ. Sapientiae, Math. 5, 93–107 (2013) 20. L. Tóth, Multiplicative Arithmetic Functions of Several Variables: A Survey, in Mathematics Without Boundaries, Surveys in Pure Mathematics, ed. by T.M. Rassias, P.M. Pardalos (Springer, New York, 2014), pp. 483–514, arXiv:1310.7053 [math.NT] 21. L. Tóth, W. Zhai, On multivariable averages of divisor functions. J. Number Theory 192, 251– 269 (2018) 22. N. Ushiroya, Mean-value theorems for multiplicative arithmetic functions of several variables. Integers 12, 989–1002 (2012) 23. J. Wu, On the primitive circle problem. Monatsh. Math. 135, 69–81 (2002)

The Rational Points Close to a Space Curve Martin N. Huxley

In memory of Eduard Wirsing

Abstract We discuss methods to find some upper bound for the number of rational points .(a/q, b/q, c/q) with least common denominator .q ≤ Q which lie close to an arc of a space curve, scaled by a factor .M ≥ 1. Keywords Rational points · Points close to a curve · Space curve

We discuss methods to find some upper bound for the number of rational points (a/q, b/q, c/q) with least common denominator .q ≤ Q which lie close to an arc of a space curve, scaled by a factor .M ≥ 1. We parametrise the space curve as .(t, F (t), G(t)), where .F (t) and .G(t) are bounded real functions, three times continuously differentiable on an open interval including [1, 2]. The derivatives .F (i) (t), .G(i) (t) for .i = 1, 2, 3 are bounded on [1, 2]. The notation is suggested by the well-known twisted cubic curve .(t, t 2 , t 3 ). Lattice point problems in number theory always require the curve or the curved boundary to be three times differentiable. We impose a twisting condition

.

.

   F (t)G (t) − F  (t)G (t) ≥ 1 for 1 ≤ t ≤ 2; B1

(1)

we use .B1 , B2 , . . . for positive constants which will appear in the calculation of O-bounds. Geometrically, the twisting condition says that the space curve does not remain close to any tangent plane.

M. N. Huxley () School of Mathematics, Cardiff University, Cardiff, Wales, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Maier et al. (eds.), Number Theory in Memory of Eduard Wirsing, https://doi.org/10.1007/978-3-031-31617-3_12

185

186

M. N. Huxley

Let f (x) = MF

.

x , M

g(x) = MG

x . M

(2)

Our criterion for the rational point .(a/q, b/q, c/q) to be close to the curve . parametrised by .(x, f (x), g(x) is    b a   .  q − f q  ≤ δ,

   c   − g a  ≤ δ. q q 

(3)

We could also take .f (x) = M  F (x/M), g(x) = M  G(x/M) in (2), and two different values .δ  and .δ  of .δ in (3). There are complications with six size parameters M, .M  , .M  , Q, .δ  and .δ  instead of the three parameters M, Q and .δ, but there are no extra difficulties. Let S be the set of rational points .(a/q, b/q, c/q) satisfying (3) with lowest denominator .q ≤ Q, and with .qM ≤ a ≤ 2qM. We want an upper bound for R, the size of the set S. If we replace the integers .a, b, c by continuous variables 2 4 .x, y, z, then the solution set has measure .O(δ MQ ); we call this the probabilistic expectation of R. The twisted cubic .(Mt, Mt 2 , MT 3 ) is a rational curve, and (3) has solutions with .δ = 0; when M is an integer, at least .Q2/3 /2 solutions .t = e/f , with .1 ≤ f ≤ Q1/3 , .f ≤ e ≤ 2f , and when .M = N 3 , the cube of an integer, at least .M 1/3 Q2/3 /2 solutions .t = e/f N with .1 ≤ f ≤ Q1/3 , .f N ≤ e ≤ 2f N. Any valid upper bound for R has to be at least δ 2 MQ4 + M 1/3 Q2/3

.

(4)

in order of magnitude. Does (4) give the best possible order of magnitude bound for R? Each of our four methods is productive for some ranges of M, Q, and .δ. They may require further conditions on the functions .F (t) and .G(t). Method 0 ( A ‘Greedy Algorithm’) For a and q fixed, there are at most .2δq + 1 values (each) for b and for c. Hence       2 .R = O (2δq + 1) = O δ 2 MQ4 + O MQ2 . q

a

We use this bound for .δ ≥ 1/6Q. Method 1 (Projection) Choose small integers k and ., not both zero, put .h(x) = kf (x) + g(x), and replace the two conditions (3) by the weaker condition .

    bk + c a   ≤ (|k| + ||)δ. − h  q q 

The Rational Points Close to a Space Curve

187

We need an extra condition that .|h (x)| is bounded away from zero. The twisting condition (1) tell us that either .|h (x)| or .|h (x)| is bounded away from zero, which is not sufficient. The good result of [6] is available  in two dimensions, but the probabilistic expectation term increases to .O δMQ2 . Method 2 (Determinants) If .δ is very small, then the .4 × 4 determinant with rows (ai , bi , ci , qi ) formed from four solutions .(ai /qi , bi /qi , ci /qi ) of (3) must be zero.

.

Method 3 (Exponential Sums) When .δ is not too small, we can detect common solutions of the two inequalities in (3) using exponential sums. Our strongest results come from Methods 0 and 3. Theorem 1.1 Let .F (t) and .G(t) be real functions three times continuously differentiable on an interval containing [1, 2], that satisfy (1). Let .M ≥ 1, .Q ≥ 1 and .δ ≤ 1 be real parameters. Let .f (x) and .g(x) be defined by (2). Then R, the number of rational points .(a/q, b/q, c/q) with least common denominator .q ≤ Q and .q ≤ a ≤ 2q that satisfy (3), is bounded by       R = O δ 2 MQ4 + O M 9/13 Q4/13 + O M 3/5 Q4/5 .

.

(5)

When .δ = 0, so the rational points satisfying (3) lie on the space curve, and .F (t) and .G(t) are four times continuously differentiable, we also have R=O

.

 √ MQ .

(6)

The results (5) and (6) of the Theorem 1.1 are closer to (4) than to the trivial bound R = O(MQ2 ). The case .δ = 0 is treated more fully in [14]. We sketch Method 2. Let .(ai /qi , bi /qi , ci /qi ), .i = 1, . . . , 4, be four solutions of (3) with the values .xi = ai /qi in a short interval of length .η. We compare the determinants

.

  a1   a2 .D =   a3  a 4

b1 b2 b3 b4

c1 c2 c3 c4

 q1  q2  , q3  q  4

  x1  x  =  2  x3 x 4

f (x1 ) f (x2 ) f (x3 ) f (x4 )

g(x1 ) g(x2 ) g(x3 ) g(x4 )

 1  1  . 1  1

In the case of rational points on a space curve, .δ = 0, and .D = q1 q2 q3 q4 . If .δ is extremely small, then D is the nearest integer to .q1 q2 q3 q4 . We require stronger conditions on the smoothness of our space curve, that .F (t) and .G(t) are four times continuously differentiable. We regard the set of four points .(xi , f (xi ), g(xi )) (in order of .xi increasing) as determining an arc of the space curve of length (measured along the x-axis) .η = x4 − x1 . When .η is small, then we use

188

M. N. Huxley

the mean value theorem repeatedly to show that  = (x2 − x1 )(x3 − x1 )(x4 − x1 )(ξ2 − ξ1 )(ξ3 − ξ1 )(ξ5 − ξ4 )

.

×(f  (ξ4 )g  (ξ6 ) − g  (ξ4 )f  (ξ6 )) for some points .ξ1 , . . . , ξ6 between .x1 and .x4 , which satisfy  2  2 η η x1 + x3 x1 + x4 , ξ2 = +O , ξ3 = +O , 2 M 2 M  2  2  2 η η η ξ1 + ξ2 ξ1 + ξ3 ξ4 + ξ5 ξ4 = +O , ξ5 = +O , ξ6 = +O . 2 M 2 M 2 M (7)

x1 + x2 +O .ξ1 = 2



η2 M



By the twisting condition (1) .

   f (ξ4 )g  (ξ6 ) − g  (ξ4 )f  (ξ6 ) ≥

 η  1 1 ≥ − O . 3 6 B1 M M 2B1 M 3

We impose a condition √ M = η0 , .η ≤ B2 Q

(8)

with .B2 chosen so large that the terms .O(η2 /M) in (7) are numerically less than .1/2Q2 . Then V , 211 B1 M 3

|| ≥

.

where V is the Vandermonde determinant, V = V (x1 , x2 , x3 , x4 ) =



.

(xj − xi ) ≥

i j >i

η3 . 4Q6

The lower bound for V comes from the observations 1 for i = 1, 2, 3, Q2

xi+1 − xi ≥

.

max(x4 − xi , xi − x1 ) ≥

η for i = 2, 3. 2

When .δ = 0 we know that .D = q1 q2 q3 q4  is a non-zero integer. Hence  1 ≤ |D| ≤ Q4 || = O

.

η 6 Q4 M3



 =O

1 B26 Q2

,

The Rational Points Close to a Space Curve

189

which is impossible for .B2 in (8) chosen sufficiently large. This contradiction shows that .x4 − x1 > η0 , a spacing property. Hence R≤

.

 √ 3M MQ . +3=O η0

(9)

When .δ is non-zero, then bi = qi f (xi ) + O(δqi ),

ci = qi g(xi ) + O(δqi ).

.

We must consider eight extra determinants with cofactors of the form ⎛

⎞ x2 g(x2 ) 1  . ⎝ x3 g(x3 ) 1 ⎠ = (x3 − x2 )(x4 − x2 )(ξ2 − ξ1 )g (ξ3 ) x4 g(x4 ) 1

(10)

for some .ξ1 , .ξ2 and .ξ3 , and twelve extra determinants with cofactors of the form  .

x3 1 x4 1

 = −(x4 − x3 ).

The relation between the determinants D and . is now   3   δη + O δ2η . .D = q1 q2 q3 q4  + O M We want the extra terms in the bracket to be less than .1/B3 Q4 for some large constant .B3 . For .η ≤ η0 we require B13 , .δ < B3 M 3/2 Q

 δ
N, but L not so large that Vinogradov’s bounds apply. The larger L is compared with N, the more derivatives have to be calculated. We use the two simplest Van der Corput bounds, which we state in the notation of (21) and (22). Van der Corput’s Second Derivative Test Let U be a subinterval of .[1, N] on which |u (x)| ≥

.

L B5 N

(24)

for some positive constant .B5 . Then  .

e(u(n)) = O



 B5 LN .

(25)

n∈U

Van der Corput’s Third Derivative Test Let V be a subinterval of .[1, N] on which |u (x)| ≥

.

L B6 N 2

(26)

for some positive constant .B6 . Then  .

e(u(n)) = O



B6 LN 4

1/6 

.

(27)

n∈V

We use the twisting condition (1) to show that intervals of types U and V cover the range for n. We have u(x) = kqMF (X) + qMG(X),

.

u(r) (x) =

where X =

x − 1 + qM , qM

 k F (r) (X) + G(r) (X). r−1 (qM) (qM)r−1

194

M. N. Huxley

√ The integers k and . are not both zero. Suppose that .|k| ≥ ||, so that .|k| ≥ L/ 2. When (24) is false, then    k   L     .  qM F (X) + qM G (X) ≤ B qM , 5 so kF  (X)G (X) + G (X)G (X) = O



.

L B5

 .

When (26) is false, then    k  L      .  q 2 M 2 F (X) + q 2 M 2 G (X) ≤ B q 2 M 2 , 6 so 









kF (X)G (X) + G (X)G (X) = O

.

L B6

 .

Subtracting and dividing by k, we have F  (X)G (X) − G (X)F  (X) = O

.



1 1 + B5 B6

 .

If .B5 and .B6 have been chosen sufficiently large, this inequality contradicts the twisting condition (1). We deduce that at each point x in .1 ≤ x ≤ N, at least one of the conditions (24) and (26) holds. When (24) holds at a point x, then by the upper bound (22) for .|u (x)|, there is a constant .B7 such that |u (y)| ≥

.

L 2B5 N

(28)

for |x − y| ≤

.

N , B5 B7

1 ≤ y ≤ N.

The interval U on which (24) holds is contained in a longer interval .U  on which (28) holds, which either meets one of the endpoints 1 or 2, or has length at least .2N/B5 B7 .

The Rational Points Close to a Space Curve

195

We partition the interval .[1, N] into at most .B5 B7 /2 + 2 disjoint intervals of type .U  , and at most .B5 B7 /2 + 1 intervals of type V . By (25) (with .B5 replaced by .2B5 ) and by (27) N  .

   B5 LN + (B6 LN 4 )1/6 e(u(n)) = O B5 B7

1

 =O

M + δ

√

qM 2/3 δ 1/6

.

Substituting in (18), we get R . =O δ4



MQ4 δ2





⎛ Q  ⎝ +O q2 · q=1

R = O δ MQ

.

2

4



1 δ4q 4

 +O

M δ



M + δ



 +O

√

⎞ qM 2/3 ⎠, δ 1/6

M 2/3 δ 1/6

 .

(29)

We see the usual paradox of the exponential sums method, that if you ask for .δ too small in (3), then the estimate (29) gets worse, not better. The third term in (29) is no bigger than the probabilistic term when δ ≥ δ1 =

.

1 . (MQ12 )2/13

The second term in (29) is no bigger than the probabilistic term when δ ≥ δ2 =

.

1 . (MQ8 )1/5

We see that .δ1 ≤ δ2 for .Q ≥ M 3/8 . For .Q ≤ M 3/8 we have .δ1 ≥ δ2 . When .δ = δ1 , then     R = O δ12 MQ4 = O M 9/13 Q4/13 .

.

(30)

The bound (30) holds in the range .δ < δ1 . For .δ > δ1 we use (3) with .δ = δ1 , but with (2) replaced by f (x) = MF

.

x + 2rδ1 , M

g(x) = MG

x + 2sδ1 M

196

M. N. Huxley

for each pair of integers r and s with  |r|, |s| ≤

.

 δ + 1. 2δ1

Then  R=O

.

δ2 2 · δ1 MQ4 δ12



  = O δ 2 MQ4 .

Similarly for .Q ≥ M 3/8 we have .δ1 ≤ δ2 . When .δ ≤ δ2 , then     R = O δ22 MQ4 = O M 3/5 Q4/5 ,

.

and for .δ ≥ δ2 we shift .f (x) by .2rδ2 and .g(x) by .2sδ2 and sum over r and s, to get  R=O

.

δ2 2 · δ2 MQ4 δ22



  = O δ 2 MQ4 .

The bound (5) of the Theorem 1.1 holds in all cases. We include references to some papers on integer or rational points close to curves or surfaces which have not been mentioned explicitly above.

References 1. E. Bombieri, J. Pila, The number of integral points on arcs and ovals. Duke Math. J. 59, 337– 357 (1989) 2. M. Branton, P. Sargos, Points entiers au voisinage d’une courbe plane à très faible courbure. Bull. Sci. Maths. (2) 118, 15–28 (1994) 3. W. Graham, G. Kolesnik, Van der Corput’s Method of Exponential Sums. London Mathematical Society Lecture Notes 126 (Cambridge University Press, Cambridge, 1991) 4. M. N. Huxley, The fractional parts of a smooth sequence. Mathematika 35, 292–296 (1988) 5. M. N. Huxley, The integer points close to a curve. Mathematika 36, 198–215 (1989) 6. M. N. Huxley, The rational points close to a curve. Ann. Scuola Norm. Sup. Pisa (Sci. Fis. Mat.) (4) 21, 357–375 (1994) 7. M. N. Huxley, Area, Lattice Points, and Exponential Sums (University Press, Oxford, 1996) 8. M. N. Huxley, The integer points close to a curve II, in Analytic Number Theory, ed. by B. C. Berndt, H. G. Diamond, A. J. Hildebrand (Birkhaüser, Basel, 1996), pp. 487–516 9. M. N. Huxley, The integer points close to a curve III, in Number Theory in Progress, ed. by K. Györy, H. Iwaniec, J. Urbanowicz (De Gruyter, Berlin, 1999), pp. 911–940 10. M. N. Huxley, The rational points close to a curve II. Acta Arith. 93, 201–219 (2000) 11. M. N. Huxley, The rational points close to a curve IV. Bonner Math. Schriften 360, 36 pp. (2003) 12. M. N. Huxley, The rational points close to a curve III. Acta Arith. 113, 15–30 (2004)

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13. M. N. Huxley, Exponential sums and the Riemann zeta function V . Proc. Lond. Math. Soc. (3) 90, 1–41 (2005) 14. M. N. Huxley, The integer points close to a space curve, in Proceedings of Conference on Diophantine Approximation, Trudy Institit Matematiki, Minsk, vol. 13 (2005), pp. 94–113 15. M. N. Huxley, The integer points in a plane curve. Funct. Approx. 37, 213–231 (2007) 16. M. N. Huxley, P. Sargos, Points entiers au voisinage d’une courbe plane de classe C n . Acta Arith. 69, 359–366 (1995) 17. M. N. Huxley, P. Sargos, Points entiers au voisinage d’une courbe plane de classe C n II. Funct. Approx. 35, 91–115 (2006) 18. M. C. Lettington, Integer points close to convex surfaces. Acta Arith. 138, 1–23 (2009) 19. O. Robert, An analogue of van der Corput’s A5 process. Mathematika 49, 167–183 (2002) 20. O. Robert, On the fourth derivative test for exponential sums. Preprint https://hal.archivesouvertes.fr/hal-01464788 21. O. Robert, P. Sargos, A fourth derivative test for exponential sums. Compos. Math. 130, 275– 292 (2002) 22. O. Robert, P. Sargos, A third derivative test for mean values of exponential sums with application to lattice point problems. Acta Arith. 106, 27–39 (2003) 23. H. P. F. Swinnerton-Dyer, The number of lattice points on a convex curve. J. Number Theory 6, 128–135 (1974)

Twists by Dirichlet Characters and Polynomial Euler Products of L-Functions, II Jerzy Kaczorowski and Alberto Perelli

In memory of Eduard Wirsing

Abstract In a previous paper we proved that if an L-function F from the Selberg class has degree 2, its conductor .qF is a prime number and F is weakly twist-regular at all primes .p = qF , then F has a polynomial Euler product. In this paper we extend this result to L-functions of degree 2 with square-free conductor .qF , which are weakly twist-regular at all primes .p  qF . Keywords Twists by Dirichlet characters · Euler products · Selberg class

1 Introduction In [5] we proved, among other things, that if an L-function F from the Selberg class S has degree 2, its conductor .qF is a prime number, and F is weakly twist-regular at all primes .p = qF , then F has a polynomial Euler product. In this paper we keep the notation from [5], but, for the reader’s convenience, we recall some basic definitions in Sect. 2. The aim of this paper is to extend the above result as follows.

.

Theorem 1.1 Let .F ∈ S be of degree 2 and its conductor .qF be square-free. If F is weakly twist-regular at all primes .p  qF , then F has a polynomial Euler product. This is not a straightforward generalization; indeed, apart from the use of Theorem 2 below, a non-trivial extension of the method in [5] is necessary. On the

J. Kaczorowski () Faculty of Mathematics and Computer Science, A. Mickiewicz University, Pozna´n, Poland Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland e-mail: [email protected] A. Perelli Dipartimento di Matematica, Università di Genova, Genova, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Maier et al. (eds.), Number Theory in Memory of Eduard Wirsing, https://doi.org/10.1007/978-3-031-31617-3_13

199

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J. Kaczorowski and A. Perelli

other hand, the present method cannot settle the problem in full generality, i.e. for all integer conductors, and some new ideas will probably be needed to prove the full result. One of the main tools in the proof is the following transformation formula for linear twists of L-functions from the extended Selberg class, which is of independent interest. Theorem 1.2 Let .F ∈ S  with .dF = 2, and let .α > 0. Then for every integer .K > 0 there exist polynomials .Q0 (s), ..., QK (s), with .Q0 (s) ≡ 1, such that F (s, α) =

.

√ −iωF∗ ( qF α)2s−1+2iθF

 K   iqF α ν ν=0

2π

  1 Qν (s)F s + ν + 2iθF , − qF α

+HK (s, α).

(1)

Here .HK (s, α) is holomorphic for .−K +

1 2

< σ < 2 and satisfies

HK (s, α)  (|s| + 1)2K+A

.

(2)

with a certain constant .A = A(F, α) > 0. Moreover, .deg Qν = 2ν and Qν (s) 

.

(A(|s| + 1))2ν ν!

for 1 ≤ ν ≤ min(|s|, K).

(3)

This should be compared with Theorem 1.2 in [4]. Apart from the value .2θF in place of .θF in (1), due to a slight change in the definition of .θF compared with [4] (see next section), the main difference is in the estimate for the size of .HK (s, α) and in the range for s in which it holds. Precisely, Theorem 1.2 in [4] states that HK (s, α)  (AK)K

.

for − K +

1 < σ < 2, |s| ≤ 2K. 2

(4)

In the proof of Theorem 1.1, we consider shifts of L-functions of the form .F (s +iτ ) with .τ → ∞ and hence reasonable control upon the size of .HK (s, α) is needed in half-planes rather than in discs. If fact, for .τ → ∞, (4) gives an estimate which is far too weak for the proof of Theorem 1.1. In contrast, (2) secures a polynomial growth which is exactly what is needed. In principle, the main structure of the proof of Theorem 1.2 is the same as that of Theorem 1.2 in [4]. Nevertheless, the new situation where s is in a half-plane forces the introduction of significant technical changes in the proof.

Twists by Dirichlet Characters and Polynomial Euler Products of L-Functions, II

201

2 Definitions Throughout the paper we write .s = σ + it, .e(x) = e2π ix and .f (s) for .f (s). The extended Selberg class .S  consists of non-identically vanishing Dirichlet series F (s) =

∞  a(n)

.

n=1

ns

absolutely convergent for .σ > 1, such that .(s − 1)m F (s) is entire of finite order for some integer .m ≥ 0, and satisfying a functional equation of type F (s)γ (s) = ωγ (1 − s)F (1 − s),

.

where .|ω| = 1 and the .γ -factor γ (s) = Q

.

s

r 

(λj s + μj )

j =1

has .Q > 0, .r ≥ 0, .λj > 0 and .(μj ) ≥ 0. Note that the conjugate function .F has conjugated coefficients .a(n). The Selberg class .S is the subclass of .S  of the functions satisfying the Ramanujan conjecture .a(n)  n and with an Euler product of the form F (s) =



.

Fp (s),

where

Fp (s) =

p

∞  a(pk ) k=0

pks

satisfies .

log Fp (s) =

∞  b(pk ) k=1

pks

with b(pk )  pϑk for a certain ϑ < 1/2.

(5)

Note that the series in (5) is absolutely convergent for .σ > ϑ and hence Fp (s) is holomorphic and bounded away from 0 for σ > ϑ for some ϑ < 1/2. (6)

.

We say that .F ∈ S has a polynomial Euler product if for every prime p there exist ∂p ∈ N and .αj,p ∈ C such that

.

Fp (s) =

∂p  

.

j =1

αj,p 1− s p

−1 .

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In such a case, it follows from the Ramanujan conjecture that .|αj,p | ≤ 1; see pp. 448–449 of [4]. The twist by a Dirichlet character .χ (mod q) is defined for .σ > 1 as F χ (s) =

∞  a(n)χ (n)

.

ns

n=1

.

We say that .F ∈ S  is weakly twist-regular at p if for every primitive Dirichlet character .χ (mod .pf ) with .1 ≤ f ≤ mqF (p), where .mqF (p) is the order of p (mod χ belongs to .S  and has the same degree as F . Moreover, the linear .qF ), the twist .F  twist of .F ∈ S is defined for .σ > 1 as F (s, α) =

∞  a(n)

.

n=1

ns

e(−nα)

with .α ∈ R. Degree .dF , conductor .qF , root number .ωF and .ξ -invariant .ξF of .F ∈ S  are defined by dF = 2

r 

qF = (2π )d Q2

λj ,

j =1 .

ωF = ω

r 

2λ

λj j ,

j =1

−2i (μj )

λj

r 

ξF = 2

,

j =1

r  (μj − 1/2) = ηF + idF θF j =1

with .ηF , θF ∈ R. We also write π

ωF∗ = ωF e−i 2 (ηF +1)

.



qF (2π )2

iθF and

 (μj )  , τF = max  j =1,...,r λj

while .mF denotes the order of the pole of F at .s = 1. In .ωF∗ , and in other definitions below, we changed .θF to .2θF compared to the corresponding definitions in [4]. This is due to the above slightly different definition of the .ξ -invariant and the fact that we are considering functions of degree .dF = 2. The H -invariants of F are defined as HF (n) = 2

.

r  Bn (μj ) j =1

λn−1 j

n = 0, 1, ...

where .Bn (x) is the n-th Bernoulli polynomial. Note that .HF (0) = dF is the degree and .HF (1) is the .ξ -invariant. We refer to our survey papers [1, 2, 6, 7] for further definitions, examples and the basic theory of the Selberg class.

Twists by Dirichlet Characters and Polynomial Euler Products of L-Functions, II

203

As in [4], for .ν, μ = 1, 2, ... we define the polynomials .Rν (s) = Rν,F (s) and Vμ (s) = Vμ,F (s) as

.

Rν (s) =Bν+1 (1 − 2s − 2iθF ) + Bν+1 (1) .

+

 ν+1   1 ν+1  (−1)ν HF (k)s ν+1−k − HF (k)(1 − s)ν+1−k k 2

(7)

k=0

and Vμ (s) = (−1)μ

.

μ  1 m!

m=1

m 



ν1 ≥1,...,νm ≥1 j =1 ν1 +...+νm =μ

Rνj (s) νj (νj + 1)

,

(8)

respectively. We also define .Q0 (s) ≡ 1 and, for .ν = 1, 2, ..., the polynomials .Qν (s) appearing in Theorem 1.2 by means of the formula exp

.

∞  (−1)ν Rν (s) ν=1

1 ν(ν + 1) (w + 2s − 1 + 2iθF )ν

≈1+

∞  ν=1

Qν (s) , (w − 1) · · · (w − ν)

and the coefficients .Cμ, , . ≥ μ ≥ 1, by

.

∞  Cμ, 1 . ≈ μ w (w − 1) · · · (w − )

(9)

=μ

Moreover, we define the coefficients .Aμ,ν (s) (.ν ≥ μ ≥ 1) by

.

∞  Aμ,ν (s) 1 . ≈ μ (w + 2s − 1 + 2iθF ) (w − 1) · · · (w − ν) ν=μ

(10)

Here .≈ means asymptotic expansion as .w → ∞, i.e. cutting the sum on the left hand side at .ν = N introduces an error of size .Os (1/|w|N +1 ). We refer to Lemmas 3.8 and 3.9 below for a more precise meaning of (9) and (10). We write .w = u + iv and, for a given s, define the contour .L (s) as follows: L (s) = L −∞ (s) ∪ L ∞ (s)

.

where L −∞ (s) = (−σ + c0 − i∞, −σ + c0 + it0 ] ∪ [−σ + c0 + it0 , −σ − c0 + it0 ]

.

L ∞ (s) = [−σ − c0 + it0 , −σ − c0 + i∞).

.

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Here .t0 = t0 (s) = c1 (|s| + 1)2 and .c0 , c1 > 0 are sufficiently large constants depending on F to be chosen later on. Observe a significant difference in the present choice of .t0 compared to the analogous choice in [4]; here .t0 is much larger, and this is important in the proof of Theorem 2. Moreover, we denote by .L ∗−∞ (s) the halfline .1 − 2s − 2iθF − L ∞ (s) taken with the positive orientation, hence L ∗−∞ (s) = (1 − σ + c0 − i∞, 1 − σ + c0 − it0∗ ]

.

with .t0∗ = t0∗ (s) = t0 + 2t + 2θF . Further, we let L ∗∞ (s) = [1 − σ + c0 − it0∗ , N + 1] ∪ [N + 1, N + 1 + i∞),

.

where the positive integer N will be chosen later on (see (44) below), and write L ∗ (s) = L ∗−∞ (s) ∪ L ∗∞ (s).

.

We shall also use the notation G(s, w) =

.

r (2π )1−r  (λj (1 − s − w) + μ¯ j )(1 − λj (s + w) − μj ) (1 − w) j =1

and S(s, w) =

.

r  2r−1  sin π(λj (s + w) + μj . sin π w j =1

Finally, .A, B, c, c , ... will denote positive constants, possibly depending on F (also via a dependence on the above constants .c0 , c1 ), not necessarily the same at each occurrence. The constants in the .- and O-symbols may also depend on .F (s) (again, also via .c0 , c1 ).

3 Lemmas Lemma 3.1 Let F ∈ S and qF be square-free. Then for σ > 1 we have  .

  ∞  q      a(n) a F −1 1 − F . e − n = F (s) dμ (s) p ns qF d

1≤a≤qF n=1 (a,qF )=1

d|qF

p|d

(11)

Twists by Dirichlet Characters and Polynomial Euler Products of L-Functions, II

205

Proof For simplicity we write q in place of qF . Recall the well-known Kluyver’s formula for the Ramanujan sum 

cq (n) :=



e (an/q) =

.

1≤a≤q (a,q)=1

μ

q  d

d|(q,n)

d.

Since cq (n) ∈ R, the left hand side of (11) equals ∞  a(n) .

n=1

ns



cq (n) =

μ

d|q

q  d

d 1−s

∞  a(dn) n=1

ns

(12)

.

Now we compute the inner sum for a generic d|q, d = p1 . . . pk . We have ∞  a(dn) .

n=1

ns

=

=

=

∞ 

∞ 



ν1 =0

νk =0

n≥1 νj pj ||n (1≤j ≤k)

∞ 

∞ 

...

...

ν1 =0

νk =0

∞ 

∞ 

...

ν1 =0

⎛ ⎝

⎛ ⎝

νk =0 k  

= F (s)d s

a(dn) ns

ν +1

k a(p j  j j =1

ν s pj j ν +1

k a(p j  j

ν s

j =1

⎞ )

pj j



⎠

(n,p1 ...pk )=1

⎞⎛ )

⎠⎝

k 

a(n) ns ⎞

Fpj (s)−1 ⎠ F (s)

j =1

 1 − Fpj (s)−1 .

j =1

Inserting this into (12) we obtain (11), and the proof is complete.

 

For F in the Selberg class S we denote by NF (σ, T ) the number of non-trivial zeros β + iγ of F in the rectangle β > σ , |γ | ≤ T . Lemma 3.2 Let F ∈ S with d = 2. Then for every > 0 and any fixed σ > 1/2 we have NF (σ, T )  T 3/2−σ + .

.

Proof See pp. 474–475 of [4].

 

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J. Kaczorowski and A. Perelli

Lemma 3.3 Let F ∈ S with d = 2. Then there exists a positive constant T0 = T0 (F ) such that  F (s) 

.

qF (2π e)2

|σ |

|s|2|σ |+1

uniformly for |t| ≥ T0 and σ ≤ −1. Proof This is a refined version of Lemma 2.1 in [4]. The proof follows from the functional equation of F and the Stirling formula; see also Lemma 3 in [5].   Lemma 3.4 Let F ∈ S , p be a prime number and σ > 1. Then there exist coefficients c(χ , p), where χ runs over the Dirichlet characters χ (mod p), such that    p −1 χ F (s). Fp (s) .F (s, 1/p) = c(χ , p)F (s) + 1 − p−1 χ ( mod p) χ =χ0

Proof See equation (2.6) of [4], observing that F (s, 1/p) = F (s, −(p −1)/p).

 

Lemma 3.5 Let F ∈ S , q be a positive integer and let θ < 1/2 be fixed. Moreover, for every prime p|q, let εp be a complex number with | p | = 1. Then there exist two positive constants a and b (depending only on θ ) and a sequence of positive numbers τk → ∞, k ≥ 1, satisfying the following two conditions (i) for every prime p|q we have |p−iτk − εp | < 1/k, (ii) |F (σ + it)| ≥ τk−b uniformly for −1 ≤ σ ≤ θ and |t − τk | ≤ τka . Proof The numbers log p, p|q are linearly independent over Q. Thus by a wellknown version of the classical Kronecker approximation theorem, for every k ≥ 1 there exists a relatively dense set of numbers τ such that |p−iτ − εp |
1 let Gk (s) :=

.

1 F (s + iτk )



F (s + iτk , a/qF ).

(19)

 q    F 1 − Fp (s + iτk )−1 d

(20)

1≤a≤qF (a,qF )=1

By Lemma 3.1 we have Gk (s) =



.

d|qF

dμ

p|d

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J. Kaczorowski and A. Perelli

hence it follows from (6) that Gk (s)  1 for σ > ϑ with a certain ϑ < 1/2.

.

(21)

As in the proof of Theorem 3 in [5], for every .a(modqF ), .(a, qF ) = 1, we fix a prime .pa ≡ a(modqF ). Obviously .F (s, a/qF ) = F (s, pa /qF ), thus we apply Theorem 2 with .s ∈ k , .α = pa /qF and .K = [|σ |] + 2 to obtain that 

 pa 2s−1+2i(τk +θF ) .F (s + iτk , a/qF ) = √ qF  K   ipa ν × Qν (s + iτk )F (s + ν + i(τk + 2θF ), −1/pa ) 2π −iωF∗

ν=0

+HK (s + iτk , pa /qF ).

(22)

Moreover, using Lemma 3.4 we have 

F (s, −1/pa ) =

.

c(χ , pa

)F χ (s) −

χ ( mod pa ) χ =χ0

 1−

 pa −1 F p (s) F (s). pa − 1 a (23)

From Theorem 2 of [5] we know that .F pa (s)−1 is a polynomial in .pa−s , and in particular is entire. Since for .s ∈ k we have that .(s + ν + i(τk + 2θF )) = 0, and the possible pole of .F (s) and .F χ (s) is at .s = 1, from (23) we deduce that .F (s + ν + i(τk + 2θF ), −1/pa ) is holomorphic for .s ∈ k . Thus from (19), (22) and (ii) of Lemma 3.5 we conclude that .Gk (s) is holomorphic for .s ∈ k . Now we estimate .Gk (s) in this region. From (19), (22) and (23) we have Gk (s) 

.

 B |σ | |Qν (s + iτk )| |F (s + iτk )| 0≤ν≤K ⎛ ⎜ × max ⎜ 1≤a≤qF ⎝ (a,qF )=1



|F χ (s + ν + i(τk + 2θF ))|

χ ( mod pa ) χ =χ0

+B |σ +ν| |F (s + ν + i(τk + 2θF ))| + max

1≤a≤qF (a,qF )=1

|HK (s + iτk , pa /qF )| , |F (s + iτk )|



(24)

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211

where here and later on .B > 1 denotes a certain constant, not necessarily the same at each occurrence. Note that we used the fact that .Fpa (s)−1 is a polynomial in .pa−s and thus it is . B |σ | . Recalling (3), Lemma 3.3 and our assumption that .F χ is a degree 2 function in .S , we see that for .σ ≤ −1 the contribution of the terms with .ν ≤ −σ − 1 is at most .

 B |σ |

(|s| + τk )2ν (|s + ν + i(τk + 2θF )| + 1)2|σ +ν|+1 0≤ν≤−σ −1 ν! (|s| + τk )2|σ |+1 max

(|σ | + τk )2ν (|σ + ν| + τk )2|σ |−2ν 0≤ν≤−σ −1 ν! (|σ | + τk )2|σ |   1 (|σ | − ν + τk ) 2(|σ |−ν)  B |σ | .  B |σ | max 0≤ν≤−σ −1 ν! |σ | + τk

 B |σ |

max

(25)

Still for .σ ≤ −1, the terms with .−σ ≤ ν ≤ K (there are at most 3 of them) contribute .

 B |σ |

τkA τkA (|σ | + τk )2|σ | |σ |  B  τkA . ([|σ |] + 1)! (|σ | + τk )2|σ | (|σ | + 1)|σ |

(26)

Finally, for .σ ≤ −1, the last term on the right hand side of (24) contributes, recalling also (2), .

|s + iτk |2|σ |+A HK (s + iτk , pa /qF )  B |σ |  B |σ | (|σ | + τk )A  B |σ | τkA . F (s + iτk ) (|σ | + τk )2|σ |+1 (27)

For .s ∈ k and .σ ≥ −1 we have .|F (s + iτk )|  τk−A whereas all the other terms in (24) are . τkA . Therefore, gathering (24)–(27) we conclude that Gk (s)  B |σ | τkA

.

for s ∈ k .

(28)

Note that the implied constant, as well as A and B, may depend on F but are independent of .τk . Let now .s ∈ k,1 . For such values we have .τkA ≤ eA|σ | ; hence, recalling our convention on the constant B, (28) gives Gk (s)  B |σ |

.

for s ∈ k,1 .

For .s ∈ k,2 we write gk (s) := exp(−4 cos(s/(2lk )))

.

and

k (s) := B s Gk (s)gk (s). G

(29)

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J. Kaczorowski and A. Perelli

k (s)  1 for s on We have .|gk (s)| ≤ 1 for .s ∈ k,2 , hence by (21) we see that .G the right vertical part of the boundary of .k,2 . Similarly, using (29) we see that the same hold for s on the left vertical part of the boundary of .k,2 . On the horizontal parts of this boundary we have .gk (s)  exp(−elk /2 ), therefore by (28) we obtain that k (s)  B σ +|σ | τkA exp(−elk /2 )  1. G

.

k (s)  1 Hence, thanks to the maximum modulus principle, we conclude that .G for .s ∈ k,2 . Suppose now that .s ∈ k,2 and .|t| ≤ 1. Since .gk (s)  k (s)  1, we have that .B s Gk (s)  1 for such exp(− exp(O(1/ lk ))  1 and .G s. This, together with (29), implies that B s Gk (s)  1

for σ ≤ 1/2 and |t| ≤ 1

.

(30)

uniformly on .τk . Now we recall that the functions .Gk (s) in (19) were defined with the help of a sequence .τk in Lemma 3.5. In particular, we have .p−iτk → εp for every .p|qF . Thus, in view of (6) and (20), for .σ > ϑ and .|t| ≤ 1 the limit G(s, ε) = lim B s Gk (s)

(31)

.

k→∞

exists and represents a bounded holomorphic function. Hence, by Vitali’s convergence theorem (see Section 5.21 of Titchmarsh [8]) the limit exists and .G(s, ε) is holomorphic for all .σ ≤ 1/2, .|t| ≤ 1. Moreover, recalling (30) and i) of Lemma 5 we have that G(s, ε)  1

(32)

.

uniformly for .ε ∈ Tr , .σ ≤ 1/2 and .|t| ≤ 1. Writing for .σ > ϑ Fp (s)−1 = 1 +

.

∞  c(pm ) , pms

m=1

it is easy to see that ∞    c(pm ) n −1 1 − Fp (s + iτk ) =− ε . .H (s, εp ) := lim k→∞ pms p m=1

Therefore, recalling (20) and (31), for .σ > ϑ we obtain G(s, ε) = B s



.

d|qF

dμ

q   F

d

p|d

H (s, εp ).

(33)

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213

 Let now .dμ(ε) be the Haar measure on .Tr = p|qF Tp , where for every .p|qF we denote by .Tp a copy of the unit circle .T. Obviously, .dμ(ε) is the product of the normalized Lebesgue measures .dμ( p ) on the circles .Tp . For a given prime .pj |qF and a positive integer m let  J (s) = J (s; pj , m) :=

.

Tr

G(s, ε)εp−m dμ(ε). j

Since .G(s, ε) is holomorphic for .σ ≤ 1/2 and .|t| ≤ 1, so is .J (s). Moreover, recalling (32), for such s we have J (s)  1.

(34)

.

From (33), Fubini’s theorem and the orthogonality relations, for .σ > ϑ we have J (s) = B s



.

dμ

d|qF

= Bs



dμ

q  



F

d

Tr

H (s, εp )εp−m dμ(ε) j

p|d

q    F

d|qF

qF = B pj μ( ) pj

d 

s

Tpj

p|d p=pj

Tp

 H (s, εp ) dμ(εp )

H (s, εpj )εp−m dμ j



εpj



Tpj

H (s, εpj )εp−m dμ(εpj ) j 

qF =B μ pj s

 c(pjm )pj1−ms .

By analytic continuation this equality holds for .σ ≤ 1/2 and .|t| ≤ 1, and using (34) we obtain m .c(pj )

pj ,m

B pjm

|σ | .

For m large enough the right hand side tends to 0 as .σ → −∞, thus .c(pjm ) = 0 for such m. This shows that .Fpj (s)−1 is a polynomial in .pj−s , and Theorem 1.1 follows.

5 Proof of Theorem 1.2 We closely follow the proof of Theorem 1.2 in [4], to which we constantly refer. So here we shall be sketchy, indicating only the main changes. In the beginning, we keep open the value of the sufficiently large constants .c0 , c1 , N below, and we add conditions when required. Moreover, we recall that the path .L (s), see Sect. 2, is defined as in Section 3.1 of [4] but with a different choice of .t0 ; this implies some

214

J. Kaczorowski and A. Perelli

differences in the estimates below compared to the analogous estimates in Section 3.3 of [4]. Let .zX = X1 + 2π iα with a large .X > 0. Writing FX (s, α) =

∞  a(n)

.

n=1

ns

exp(−nzX ),

as in (3.27) of [4] for .σ < 2 we have   1 1 −w −w .FX (s, α) = F (s + w)(w)zX dw = F (s + w)(w)zX dw. 2π i (2−σ ) 2π i L (s) (35) If .w = u + iv ∈ L −∞ (s) then .(s + w) ≥ −c0 , hence .F (s + w)  |s + w|c for some .c > 0 since .F (s) has polynomial growth on vertical strips. If in addition .v < t0 , then .(s + w) = c0 > 1 and hence .F (s + w)  1. Moreover, still for .w ∈ L −∞ (s), we have −w |zX | = |zX |−u exp(v(π/2 − η(X))),

.

where .η(X) > 0 and .η(X) = O(1/X), and by Stirling’s formula π

1

(w)  e− 2 |v| (|v| + 1)u− 2 .

.

Therefore, due to the different choice of .t0 , the contribution of .L −∞ (s) to (35) is .

1 2π i

 L −∞ (s)

−w F (s + w)(w)zX dw  A|σ | (|s| + 1)2|σ |+c

for some positive A and c; such constants will not necessarily be the same at each occurrence. As a consequence, for .σ < 2 and any fixed .α > 0, equation (3.28) of [4] becomes  1 −w FX (s, α) = F (s + w)(w)zX dw + O(A|σ | (|s| + 1)2|σ |+c ) 2π i L (s) ∞ . (36) = I X (s, α) + O(A|σ | (|s| + 1)2|σ |+c ), say, uniformly as .X → ∞. As on p. 467 of [4], now we apply the functional equation of .F (s) and the reflection formula of .(s), thus getting I X (s, α) = ωQ

.

1−2s

1 2π i

 L ∞ (s)

F (1 − s − w)G(s, w)S(s, w)(Q2 zX )−w dw;

Twists by Dirichlet Characters and Polynomial Euler Products of L-Functions, II

215

see Sect. 2 for definitions. Taking into account the different choice of .t0 , following Lemmas 3.1 and 3.2 of [4] for .w ∈ L ∞ we have S(s, w) = −ie(−ξF /4)e−π is (1 + O(e−ηv ))

.

for a certain positive .η and π

G(s, w)  e− 2 (v+2t) (v + |s| + 1)2|σ |+c .

.

Thus replacing .S(s, w) by .−ie(−ξF /4)e−π is we obtain I X (s, α) = −iωe(−ξF /4)Q1−2s e−π is  1 F (1 − s − w)G(s, w)(Q2 zX )−w dw 2π i L ∞ (s)  ∞ |G(s, −σ − c0 + iv)||(Q2 zX )σ +c0 −iv ||e−π is |e−ηv dv) +O(A|σ |

.

t0

= J X (s, α) + O(A|σ | (|s| + 1)2|σ |+c ),

(37)

say, uniformly as .X → ∞. As on p. 467 of [4], we reduce .G(s, w) in (37) to a single .-factor by means of the uniform version of the Stirling formula in [3]. For .1 ≤ N ≤ |s| + c and r 2λj .β = j =1 λj , arguing as in [4] we obtain  −2i μj 1 log G(s, w) = log (1 − 2s − w − 2iθF ) + ( − s − w) log β + log λj 2 r

j =1

.

+

  N  Rν (s) 1 (c(|s| + 1))N +2 . + O ν(ν + 1) w ν |w|N +1 ν=1

(38)   N+2 < 1 for .w ∈ L ∞ (s), provided the constants in the Since .O (c(|s|+1)) |w|N+1 definition of .L ∞ (s) are sufficiently large, for .1 ≤ N ≤ |σ | + c we have 

e

.

O

(c(|s|+1))N+2 |w|N+1



  (|s| + 1)N +2 . = 1 + O A|σ | |w|N +1

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J. Kaczorowski and A. Perelli

Hence from (38), we obtain G(s, w) =(1 − 2s − w − 2iθF )β

1 2 −s−w

r  j =1

.

−2i μj λj



N  Rν (s) 1 exp ν(ν + 1) w ν



ν=1

  N +2  |σ | (|s| + 1) . × 1+O A |w|N +1 (39)

Moreover, using Lemma 3.6, for .w ∈ L ∞ (s) and .1 ≤ N ≤ c(|s| + 1) we have N N   Rν (s) 1 (c (|s| + 1))ν+1 1  ν(ν + 1) w ν ν(ν + 1) |w|ν ν=1

ν=1

.



N  ν=1

 (c (|s| + 1))ν+1 1 1 c = ( )2ν  1 2ν ν(ν + 1) (c1 (|s| + 1)) ν(ν + 1) c1 N

ν=1

(40) if .c1 ≥ c . Again, we remark that (39) and (40) hold thanks to our present choice of .t0 in the definition of .L ∞ (s). This small but significant change compared to [4] leads to much better estimates in the s-aspect (compare to Lemma 3.9 and (3.31) in [4]). Next we replace .G(s, w) by its main term in (39) inside the integral .J X (s, α) in (37). This causes an error of the size  A|σ | eπ t |(1 − 2s − w − 2iθF )|

L ∞ (s)

N

  N +2  Rν (s) 1  2 −w  (|s| + 1) × exp  z ) |dw|  (Q X ν N +1 ν(ν + 1) w |w| ν=1 .   |dw|    |σ | π t N +2 A e (|s| + 1) |(1 − 2s − w − 2iθF )| (Q2 zX )−w  |w|N +1 L ∞ (s)  ∞ A|σ | (|s| + 1)N +2 e−π v/2 eπ v/2 v −σ +c0 −N −1/2 dv  A|σ | (|s| + 1)N +2 t0

if .N ≥ −σ + c0 + 1, the bound being uniform in X. Indeed, the bound |(1 − 2s − w − 2iθF )(Q2 zX )−w ||w|−N −1  A|σ | e−π t v −σ +c0 −N −1/2 ,

.

Twists by Dirichlet Characters and Polynomial Euler Products of L-Functions, II

217

used to obtain the last estimate, follows by an application of Stirling’s formula, namely (1 − 2s − w − 2iθF )  |1 − 2s − w − 2iθF |1/2−σ +c0 × exp((2t + v + 2θF ) arg(1 − 2s − w − 2iθF ))

.

 v 1/2−σ +c0 exp((2t + v + 2θF ) arg(1−2s−w−2iθF )), observing that 

2t + v + 2θF . arg(1−2s −w−2iθF ) = − arctan 1 − 2σ − u



π = − +O 2



|σ | + 1 2t + v + 2θF

 .

Therefore, from (37), (39) and recalling the definition of .qF and .ωF∗ , for |σ | + c ≤ N ≤ |σ | + c + 1

.

(41)

we have J X (s, α)

.

  q 1/2−s−iθF 1 1 F e(− (s + iθF )) F (1 − s − w) 2 2π i L ∞ (s) 4π 2 N

 Rν (s) 1  qF zX −w × (1 − 2s − w − 2iθF ) exp dw ν(ν + 1) w ν 4π 2 ν=1   + O A|σ | (|s| + 1)2|σ |+c .

=ωF∗

Hence, by the substitution .1−2s −w−2iθF → w in the above integral and recalling the definition of .L ∗−∞ (s) in Sect. 2 we obtain

J X (s, α) =

.

−iωF∗

 √ 2s−1+2iθF qF √ qF α − i 2π X

 1 F¯ (s + w + 2iθF )(w) 2π i L ∗−∞ (s) N

 q z w  (−1)ν Rν (s) 1 F X dw × exp ν ν(ν + 1) (w + 2s − 1 + 2iθF ) 4π 2 ν=1   + O A|σ | (|s| + 1)2|σ |+c ) , (42) uniformly as .X → ∞.

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Writing again .η = 2s − 1 + 2iθF , from the power series expansion of the exponential function and recalling (13) we have N

∞  (−1)ν Rν (s)  Vμ,N (s) 1 exp =1+ ν(ν + 1) (w + η)ν (w + η)μ ν=1

μ=1

⎛ ⎞ N ∞   |Vμ,N (s)| Vμ (s) ⎠. + O⎝ =1+ (w + η)μ |w + η|μ

.

μ=N +1

μ=1

(43) Since for .w ∈ L ∗−∞ (s) we have .|w + η| ≥ 2(c (|s| + 1))2 , where .c denotes the constant in Lemma 3.7, using such a lemma we obtain ∞ ∞ ∞   |Vμ,N (s)| (c (|s| + 1)2μ (c (|s| + 1)2(N +1)  −μ )  ≤ 2 |w + η|μ |w + η|μ |w + η|N +1

.

μ=N +1

μ=N +1

 A|σ |

μ=N +1

(|s| + 1)2N +1 . |w(w − 1) · · · (w − N )|

Thus using (43) and Lemmas 3.7 and 3.9

.

exp

N  (−1)ν Rν (s)

1 ν(ν + 1) (w + η)ν

ν=1

=1+

N  μ=1

1+

N 



  Vμ (s) (|s| + 1)2N +1 |σ | + O A (w + η)μ |w(w − 1) · · · (w − N)| Vμ (s)

μ=1

⎛

N 

Aμ,ν (s) (w − 1) · · · (w − ν) ν=μ

⎞ N N N!  4 +O ⎝ |Vμ (s)||η|N −μ+1 ⎠ |w(w − 1) · · · (w − N)| μ=1

 +O A|σ | =

(|s| + 1)2N +1 |w(w − 1) · · · (w − N)|



N 

Qν (s) (w − 1) · · · (w − ν) ν=0 ⎛ ⎞ N N N μ+1  4 N!(c (|s| + 1) (c (|s| + 1) ⎠ +O ⎝ |w(w − 1) · · · (w − N)| (μ − 1)! μ=1

Twists by Dirichlet Characters and Polynomial Euler Products of L-Functions, II

219

 (|s| + 1)2N +1 |w(w − 1) · · · (w − N)|   N  Qν (s) (|s| + 1)2N +1 |σ | = +O A (w − 1) · · · (w − ν) |w(w − 1) · · · (w − N )|  +O A|σ |

ν=0

 Now we replace the term .exp( N ν=1 ...) in (5) by the main term of the above formula. This causes a further error of size  |(w)| |σ | 2N +1 eπ |v|/2 |dw|  A (|s| + 1) ∗ L −∞ (s) |w(w − 1) · · · (w − N)| .  |(w − N)|eπ |v|/2 |dw|. = A|σ | (|s| + 1)2N +1 |w| L ∗−∞ (s) Moreover, we also want N such that .(w − N) ≤ 0 for .w ∈ L ∗−∞ (s), i.e. we choose N = [−σ ] + k

.

(44)

with a sufficiently large positive integer k satisfying (41). With such a choice of N we have .|(w − N)|  e−π |v|/2 /|v|1/2 , hence the integral is . 1. Therefore (5) becomes, uniformly as .X → ∞,  √ 2s−1+2iθF qF √ qF α − i J X (s, α) = − iωF∗ 2π X  N  q z w  1 F X . F (s + w + 2iθF )(w − ν) dw × Qν (s) 2π i L ∗−∞ (s) 4π 2 ν=0   + O A|σ | (|s| + 1)2|σ |+B , (45) which is the analog of (3.35) in [4]. Replacing the path of integration in (45) by the whole path .L ∗ (s) causes an error which, since .ν ≤ N, by Lemma 3.10 is of size .

|σ |

A

 N  (|s| + 1)2ν ν=0

ν!

L ∗∞ (s)

|F (s + w + 2iθF )(w − ν)

 q z w F X ||dw|. 4π 2

Let now .w ∈ L ∗∞ (s). Clearly, .F (s + w + 2iθF )  1. Moreover, for .0 ≤ ν ≤ N we have .−c ≤ (w − ν) ≤ N + 1 − ν and hence for .v = 0   π ν−u = |v| + O(|σ | + 1). .v arg(w − ν) = v arg(iv) + arg 1 + i v 2

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Observing that this estimate holds for .v = 0 as well, by Stirling’s formula we get .

log |(w − ν)| = (u − ν − 1/2) log |w − ν| −

π |v| + O(|σ | + 1). 2

Consequently π

(w − ν)  A|σ | e− 2 |v| (|s| + |v| + 1)N −ν+1/2 .

.

w |  A|σ | e−v arg zX , thus the above mentioned error is (with a suitable Further, .|zX .c > 0)

.

 A|σ | 

N  (|s| + 1)2ν

ν!

ν=0 ∞ −t0∗ (s)



(|s| + |v| + 1)|σ |−ν+c e−π |v|/2−v arg zX dv  A|σ | (|s| + 1)2|σ |+B .

We also have by Cauchy’s theorem that for .0 ≤ ν ≤ N 

 q z w F X dw 4π 2 L ∗ (s)  |σ |+ν+2+i∞  q z w 1 F X F (s + w + 2iθF )(w − ν) dw =. 2π i |σ |+ν+2−i∞ 4π 2   ∞  q qF α ν  a(n) 4π 2 F + i exp − n = 2π qF zX 4π 2 X ns+ν+2iθF 1 2π i

F (s + w + 2iθF )(w − ν)

n=1

since the poles of the integrand lie to the left of .L ∗ (s). Consequently, (45) becomes J X (s, α) = − iωF∗ .

×

∞  n=1

 √

√

qF α − i

a(n) ns+ν+2iθF

qF 2π X

2s−1+2iθF  N  qF α ν qF + i Qν (s) 2π 4π 2 X ν=0

  4π 2 exp − n + O(A|σ | (|s| + 1)2|σ |+B ), qF zX (46)

uniformly as .X → ∞. Note that, since .

−

1 1 2π i 4π 2 − , = qF zX qF α qF α 2 X + O(1)

the series in (46) is absolutely convergent for all s, for every .ν.

Twists by Dirichlet Characters and Polynomial Euler Products of L-Functions, II

221

As on p. 472 of [4], the final step is to make the range of summation of .ν in (46) independent of .σ (recall that N depends on .σ , see (44)). Let .K > 0 be a large integer and .σ > −K + 1/2. Depending on the relative sizes of N and K, we add to or withdraw from (46) the terms with .ν between .N + 1 and K or between .K + 1 and N, respectively. In both cases we have that .σ + ν > 3/2 for such .ν’s (call them .ν ∈ X ), hence from Lemma 3.10 we deduce that −iωF∗

 √

√ 2s−1+2iθF   qF qF α ν qF + i Qν (s) qF α − i 2π X 2π 4π 2 X ν∈X

.

×

∞  n=1



  (c (|s| + 1))2ν 4π 2 exp − n  A|σ | qF zX ν!

a(n) ns+ν+2iθF

(47)

ν∈X

uniformly in X. If .N < K this is .

 A|σ | (|s| + 1)2K

 c 2ν  c 2ν A|σ | c 2N  (|s| + 1)2K  (|s| + 1)2K , ν! (N + 1)! ν!

ν≥N +1

ν≥0

while if .N > K we have A|σ |

.

 (c (|s| + 1))2ν (c (|s| + 1)2N  A|σ |  (|s| + 1)2N  (|s| + 1)2K+A . ν! (N − 1)!

ν∈X

Thus, in view of (36), (37), (46) and (47), for .−K + 1/2 < σ < 2 we have FX (s, α) .

= − iωF∗

 √ 2s−1+2iθF  K  qF qF α ν qF √ + i qF α − i 2π X 2π 4π 2 X ν=0

(48)

× Qν (s)FX∗ (s + ν + 2iθF , α) + HX (s, α), where ∗ .FX (s, α)

=

∞  a(n) n=1

ns



 4π 2 exp − n qF zX

and HX (s, α)  (|s| + 1)2K+A

.

uniformly as .X → ∞. Moreover, since .FX (s, α), .FX∗ (s, α) and .Qν (s) are entire functions, .HX (s, α) is also entire. Further, from (48) we have that for .1 < σ < 2 .

lim HX (s, α) = H (s, α)

X→∞

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exists and is holomorphic since this is clearly true for .FX (s, α) and .FX∗ (s, α). Thanks to (48), for .1 < σ < 2 we also have that H (s, α) = F (s, α) + iωF∗

.

√

qF α

2s−1+2iθF

  K   qF α ν 1 ¯ . i × Qν (s)F s + ν + 2iθF , − 2π qF α

(49)

ν=0

Hence by Vitali’s convergence theorem the limit function .H (s, α) exists and is holomorphic for .−K + 1/2 < σ < 2, and satisfies H (s, α)  (|s| + 1)2K+A .

.

This provides analytic continuation and bounds for the right-hand side of (49), and Theorem 2 follows. Acknowledgments We wish to warmly thank the referee for carefully reading our manuscript and for pointing out many inaccuracies. This research was partially supported by the Istituto Nazionale di Alta Matematica, by the MIUR grant PRIN-2017 “Geometric, algebraic and analytic methods in arithmetic” and by grant 2021/41/BST1/00241 “Analytic methods in number theory” from the National Science Centre, Poland.

References 1. J. Kaczorowski, Axiomatic theory of L-functions: the Selberg class, in Analytic Number Theory, C.I.M.E. Summer School, Cetraro, 2002, ed. by A. Perelli, C. Viola. Springer L.N. 1891 (2006), pp. 133–209 2. J. Kaczorowski, A. Perelli, The Selberg class: a survey, in Number Theory in Progress, Proceedings of Conference in Honor of A. Schinzel, ed. by K. Györy et al. (de Gruyter, Berlin, 1999), pp. 953–992 3. J. Kaczorowski, A. Perelli, A uniform version of Stirling’s formula. Funct. Approx. 45, 89–96 (2011) 4. J. Kaczorowski, A. Perelli, Twists, Euler products and a converse theorem for L-functions of degree 2. Ann. Scuola Norm. Sup. Pisa (V) 14, 441–480 (2015) 5. J. Kaczorowski, A. Perelli, Twists by Dirichlet characters and polynomial Euler products of L-functions. https://doi.org/10.48550/arXiv:2303.02417 6. A. Perelli, A survey of the Selberg class of L-functions, part II. Riv. Mat. Univ. Parma (7) 3*, 83–118 (2004) 7. A. Perelli, A survey of the Selberg class of L-functions, part I. Milan J. Math. 73, 19–52 (2005) 8. E.C. Titchmarsh, Theory of Functions (Oxford University Press, London, 1952)

Solving the Iterative Differential Equation −γ g  = g −1 .

Roland Miyamoto and Jürgen Sander

Abstract We solve the iterative differential equation .−γ g  = g −1 for .g : [0, 1] → [0, 1] and for one particular parameter .γ = κ ≈ 0.27887706137 and show that it √ has no solution for larger parameters, nor for positive parameters .γ < 3−6 3 . Keywords Iterative differential equation · Operator · Fixed point · Convergence theorem

1 Introduction The problem we are addressing in this paper naturally arises from the study of Levine’s sequence [8] but is interesting in its own right as it can be understood with high-school knowledge of calculus and is described nicely in geometric terms. To come to the point, we are looking for a differentiable function .g : [0, 1] → [0, 1] with the following property: When we rotate (the graph of) g clockwise by .90◦ about the origin and subsequently stretch it vertically by a suitable positive factor, then we obtain its derivative .g  . A function g with this property will obviously be continuously differentiable and strictly decreasing with .g(0) = 1 and .g(1) = 0. Moreover the “suitable factor”

R. Miyamoto () Hildesheim, Germany e-mail: [email protected] J. Sander Institut für Mathematik und Angewandte Informatik, Hildesheim, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Maier et al. (eds.), Number Theory in Memory of Eduard Wirsing, https://doi.org/10.1007/978-3-031-31617-3_14

223

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R. Miyamoto and J. Sander

1 must equal the reciprocal of the area .γ := 0 g that g encloses with the axes, so that 1  .g (0) = − . Formally speaking, g satisfies the iterative differential equation (IDE) γ .

− γ g  = g −1

(Dγ )

where .g −1 denotes the compositional inverse of g. Similar IDEs have been studied by Eder [3], Feˇckan [5], Buic˘a [2], Egri and Rus [4] and Berinde [1], but the techniques employed there seem not to help in the situation at hand. The only solution to our problem that we were able to find is depicted on the right, and we believe that there is no other. We have produced this solution numerically by an iterative process. At each step of this process, we simply perform the said operation, denoted T from now on: Given a decreasing function 1 ◦ .h1 : [0, 1] → [0, 1] with non-zero area .γ1 := 0 h1 , we rotate it by .90 about the origin, then stretch it vertically by . γ11 , then integrate, to obtain the next function .h2 := T h1 : [0, 1] → [0, 1]. Starting from the straight line given by .h1 (x) = 1 − x, the sequence of iterates .h1 , h2 := T h1 , h3 := T h2 , . . . provably converges to a function h which solves the IDE (D.κ ) where .κ = h ≈ 0.27887706137. Moreover, we show that any function solving (Dγ ) for some .γ > 0 is bounded by h from above. We want to briefly indicate the connection with Levine’s integer sequence [8] (ln )n∈N = (1, 2, 2, 3, 4, 7, 14, 42, 213, 2837, 175450, 139759600, . . .).

.

Levine defines .ln as the largest summand in a partition .λn  ln+2 with .ln+1 summands, while recursively constructing .λn from .λn−1 . Mallows [11, p. 152f] observed that (the Young diagrams of) the partitions .λn resp. their duals .λ∗n , when (rotated anticlockwise by .90◦ and) scaled down into the unit square .[0, 1] × [0, 1], seem to converge to limit functions .g −1 resp. g satisfying the IDE (Dγ ) with 1  1 −1 .γ = 0 g = 0 g , and he obtained the estimate .γ ≈ 0.277 by developing the supposed limit function .g −1 into a power series (cf. [10], [11, pp. 151–155] and [9]).

Solving the Iterative Differential Equation .−γ g  = g −1

225

1

2

2

0

1 2

−1 1 2

−2

−3

−1

However, the existence of Mallows’ functions g and .g −1 , i.e. the convergence, has not been proven until today, while it would allow for a better asymptotic estimate than the rather coarse one .

log ln ∼ c · ϕ n

with ϕ =

√ 1+ 5 2

and

c ≈ 0.05427

found by Poonen and Rains [10]. The question of to which extent the results presented here alleviate the open tasks for Levine’s sequence will be explored in a separate paper.

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2 The Operator T For .0 ≤ a ≤ b ≤ 1 and any (Lebesgue) measurable function .f : [0, 1] → [0, ∞), b  1 b we abbreviate . a f := a f (x)dx and . f := 0 f . We will also conveniently write .id := id[0,1] for the identity function on .[0, 1]. Our investigations will involve the spaces .M

:= {f : [0, 1] → [0, ∞) : f measurable,



f > 0},

E := {f ∈ M : f decreasing, f (0) = 1}, C := {g ∈ E : g continuous, g(1) = 0},

C˘ := {g ∈ C : g convex},

D := {g ∈ C : g strictly decreasing},

˘ := D ∩ C˘ , D

D  := {f ∈ D : f continuously differentiable on (0, 1]},

˘  := D  ∩ C˘ , D

D ` := {g ∈ D  : g  (1) = 0, lim g  (x) ∈ (−∞, 0] exists},

˘ ` := D ` ∩ C˘ D

x→0

of functions on .[0, 1]. Given .g ∈ E , we will use its pseudo-inverse .g ∗ ∈ E defined by .g

∗

(y) := sup g −1 [y, 1] = sup{x ∈ [0, 1] : g(x) ≥ y}

for y ∈ [0, 1].

Among others, Klement et. al. [7] and Feng et. al. [6] have studied generalised inverses of increasing functions. By [6], our definition is compatible in that .1 − g ∗ equals the pseudo-inverse of .g ◦ (1 − id) in the sense of [7]. According to [7] and Remark 2.1(c) below, .g ∗ equals the compositional inverse .g −1 if .g ∈ D . Thus we may, and shall, consistently write .g ∗ in all cases from now on. Remark 2.1 For .f, g ∈ E , the following statements hold. (a) (b) (c) (d) (e) (f) (g) (h)

≤ g ⇒ f ∗ ≤ g ∗ . If .g ∈ C , then .g ∗ is strictly decreasing. If .g ∈ D , then .g ∗ = g −1 ∈ D is the inverse function of g. ˘ ⇐⇒ g ∗ ∈ D ˘. .g ∈ D ˘ ` ⇒ g ∗ ∈ D ˘ . .g ∈ D   ∗ . g = g .   . |f − g| = |f ∗ − g ∗ |. 1 ∗ t . g(t) g = 0 g − t · g(t) for all .t ∈ [0, 1]. .f

Proof (a) Let .y ∈ [0, 1] and suppose that .f ≤ g. Then .f −1 [y, 1] ⊆ g −1 [y, 1], hence .f ∗ (y) ≤ g ∗ (y). (b) .0 ≤ y1 ≤ y2 ≤ 1 implies .g −1 [y1 , 1] ⊇ g −1 [y2 , 1], hence .x1 := g ∗ (y1 ) ≥ x2 := g ∗ (y2 ) and .x1 = x2 ⇒ g −1 [y1 , y2 ) = ∅ ⇒ y1 = y2 by the intermediate value theorem for .g ∈ C .

Solving the Iterative Differential Equation .−γ g  = g −1

227

(c) If .g ∈ D , then .g −1 ∈ D and .g ∗ (y) = sup g −1 [y, 1] = g −1 (y) for all .y ∈ [0, 1]. ˘ , .0 ≤ x1 < x2 ≤ 1 and .t ∈ [0, 1]. Then .y1 := g(x1 ) > y2 := (d) Suppose that .g ∈ D g(x2 ), .g ∗ = g −1 ∈ D , .x1 = g ∗ (y1 ) and .x2 = g ∗ (y2 ) by (c) and .ty1 + (1 − t)y2 ≥ g(tx1 +(1−t)x2 ), hence .g ∗ (ty1 +(1−t)y2 ) ≤ g ∗ (g(tx1 +(1−t)x2 )) = tx1 +(1−t)x2 . ˘ `. Then .a := g  (0) := limx→0 g  (x) ≤ 0, .g  (1) = 0 and .g  : [0, 1] → [a, 0] (e) Let .g ∈ D is increasing, hence .g  (x) = 0 ⇒ g  |[x,1] = 0 ⇒ g|[x,1] = g(1) = 0 ⇒ x = 1 for any .x ∈ [0, 1]. Therefore .g  ([0, 1)) = [a, 0) and .a < 0, so that .g ∗ = g −1 is continuously differentiable on .g([0, 1)) = (0, 1]. (f) To g we associate its area set .Ag := {(x, y) ∈ [0, 1]2 : g(x) ≥ y}, with topological closure .A¯ g and its set .Jg := {x ∈ [0, 1] : g(x) < inf g[0, x)} of left discontinuities. Then .A¯ g ∗ = A¯ g ⊆ Ag ∪ (Jg × [0, 1]). Let .λ2 be the Lebesgue measure on .[0, 1]2 . Because g is  decreasing and bounded, it has (at most) countably many discontinuities. Therefore, . g = λ2 (Ag ) = λ2 (A¯ g ) = λ2 (A¯ g ∗ ) = λ2 (Ag ∗ ) = g ∗ . (g) The two functions .u := min{f, g} and .v := max{f, g} satisfy .u, v

∈ E,

u∗ = min{f ∗ , g ∗ },

v ∗ = max{f ∗ , g ∗ }

      and therefore . |f − g| = v − u = v ∗ − u∗ = |f ∗ − g ∗ | by (f). (h) Let .t ∈ [0, 1] and define .h ∈ E by setting .h(x) := g(x) for .x ∈ [0, t] and .h(x) := 0 ∗ t for .y ∈ (0, g(t)] and .h∗ (y) = g ∗ (y) for .y > g(t), hence for  t .x >t. Then  .h∗ (y)= g(t) ∗ 1 1 . h + g(t) h∗ = t · g(t) + g(t) g ∗ by (f). 0 g = h= h = 0   ˘ , we define the continuous functions If and Dh by For given .f ∈ M and .h ∈ D` setting .(If )(x)

1 f := x f

and

(Dh)(x) :=

h (x) h (0)

for x ∈ [0, 1]

and formally introduce the operator .T : E → C described in the introduction by setting .T g

∗

:= Ig for g ∈ E ,

that is,

1 ∗ g for x ∈ [0, 1] (T g)(x) = x g

by Remark 2.1(f), as well as its iterations .T 0 = idE , .T 1 = T , .T n+1 := T ◦ T n for .n ∈ N. Proposition 2.2 For .g ∈ C and .f ∈ D , the following statements hold. ˘. (a) .T g ∈ D  t (b) .(T g)(g(t)) · g = 0 g − tg(t) for all .t ∈ [0, 1]. ˘ ` with .(If ) (0) = −  1 and .DIf = f . (c) .If ∈ D f

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R. Miyamoto and J. Sander

Proof (a) Let .0 ≤ a < b ≤ 1. Then .g ∗ (a) > g ∗ (b) by Remark 2.1(b), hence 

  b g · (T g)(a) − (T g)(b) = a g ∗ > (b − a) · g ∗ (b) ≥ 0,

.

showing that T g is strictly decreasing. Similarly, for .a < x < b, we obtain  .

g·

x ∗ b ∗ g g (T g)(a) − (T g)(x) = a > g ∗ (x) > x x−a x−a b−x  (T g)(x) − (T g)(b) , = g· b−x

˘. hence T g is convex. In total, we have proved .T g ∈ D (b) follows from Remark 2.1(f) and (h). (c) .(If ) = − ff is continuous and increasing, .(If ) (0) = −  1f and .(If ) (1) = 0, so ˘ ` and .DIf = that .If ∈ D

(If ) (If ) (0)

= f.

 

We now explicitly state the connection between the operator T and the IDE (Dγ ). Theorem 2.3 A function .g ∈ C is a fixed point of the operator T if and only if g solves the IDE (Dγ ) for some .γ > 0, and then g also satisfies the following properties: ˘  ∈ D `. . g = γ. 1  .g (0) = − . γ ∗  .g and .g are continuously differentiable on the interval .(0, 1].  ∗  .g (1) = 1 and .(g ) (1) = −γ .  Proof First we assume that .g = T g ∈ C and set .α := g. Using Proposition 2.2(a), ˘ and then .g ∈ D ˘ ` by Remark 2.1(c) and Proposition 2.2(c), we conclude that .g ∈ D 1 settling assertion (a). Differentiating the equation .g(x) = (T g)(x) = α1 x g ∗ , we arrive at .−αg  = g ∗ , that is, g solves (D.α ). Conversely assume that .g ∈ C (is differentiable and) solves (Dγ ) for some .γ > 0. 1 Integrating (Dγ ) while considering Remark 2.1(c) leads to .g(x) = γ1 x g ∗ . Plugging 0  into this, yields .1 = g(0) = γ1 g by Remark 2.1(f), thereby showing (b) and .g = T g.

(a) (b) (c) (d) (e)

.g

∗

(c) Plugging 0 into (Dγ ) and using Remark 2.1(c) gives .g  (0) = −gγ (0) = −1 γ . (d) By (a) and Remark 2.1(e), we have .g ∗ ∈ D  , and the assertion follows from (Dγ ). (e) Plugging 1 into the derivative of (Dγ ), yields .−γ g  (1) = (g ∗ ) (g(0)) = g 1(0) = −γ by the chain rule and (c), thus .g  (1) = 1 and .(g ∗ ) (1) = −γ .  

By Theorem 2.3(a), every solution to (Dγ ) necessarily lies in .D˘ `. Thus .

 ˘ `} = { g : g ∈ C , g = T g} := {γ > 0 : (Dγ ) has a solution g ∈ D

Solving the Iterative Differential Equation .−γ g  = g −1

229

is the set of parameters .γ > 0 for which (Dγ ) has a solution. Corollary 2.4 . ⊆ (0, 12 ).

 ˘ ` be a solution to (Dγ ). Then .γ = g ∈ (0, 1 ] by Proof Let .γ > 0 and .g ∈ D 2   Theorem 2.3(b). But .γ = 12 means that .g = 1 − id, which does not solve (D. 1 ). 2

The following lemma will help us narrow down the set . further. Lemma 2.5 Let .g ∈ C˘ , .γ := Then



g, .β := inf g −1 {0} and .α ∈ (0, 1] such that .g ≥ 1 −

id α.

 (a) .α ≤ 2γ ≤ β ≤ 1, and .α = 2γ ⇒ 2γ = β ⇒ T g = 13 , (b) .(T g) : [0, 1] → (−∞, 0] exists, is continuous, strictly increasing and concave, ˘ ` with .(T g) (0) = − β , (c) .T g ∈ D γ (d)

.

β

γ

(id − 1) ≤ (T g) ≤

α γ (id − 1),

(e) . T g ≤ 13 ,  (f) .αβ − 4αγ + 4γ 2 ≤ 6(β − α)γ · T g. Proof (a) From .g ∈ C˘ and the definition of .β, we infer that .g(x) ≤ 1 − βx for all .x ∈ [0, β], α  β β id hence . α2 = 0 (1 − id α ) ≤ g = γ ≤ 0 (1 − β ) = 2 , settling the asserted inequality α chain. From this, we also see that . 2 = γ ⇐⇒ g|[α,1] = 0 ⇒ β = α and that  β ∗ 2 T g = 13 . .γ = 2 ⇒ g (y) = β(1 − y) for .y ∈ (0, 1] ⇒ T g = (1 − id) ⇒ (b) By its convexity, g is strictly decreasing on .[0, β]. Thus .f (x) := g(βx) for .x ∈ [0, 1] ˘ , which satisfies .1− β id ≤ f ≤ 1−id. Using Remark 2.1(d), defines a function .f ∈ D α ˘, (a) and (f), we infer that .f ∗ ∈ D .

α β (1 − id)

≤ f ∗ ≤ 1 − id and



f∗ =



f =

1 β



g=

γ β.

(1)

Because .βf ∗ (x) = g ∗ (x) for all .x ∈ (0, 1], we conclude that .T g = Tf is differentiable with continuous derivative 

.(T g)

= (Tf ) = − γβ f ∗ ,

(2)

and the assertions follow. (c) From (2) we infer that .(T g) (0) = − γβ and .(T g) (1) = 0, hence .T g ∈ D `, while ˘ holds by Proposition 2.2(a). .T g ∈ D (d) follows from (1) and (2).  (e) By (a)–(c) and because . (T g) = (T g)(1) − (T g)(0) = −1 = (2id − 2), .s

:= sup{0 < x < 1 : (T g) (x) ≤ 2x − 2} ∈ (0, 1]

is well-defined, .(T g) |[0,s] ≤ 2id[0,s] − 2 and .(T g) |[s,1] ≥ 2id[s,1] − 2. We conclude x that .(T g)(x) ≤ 1 + 0 (2id − 2) = (1 − x)2 for .x ∈ [0, s] and also .(T g)(x) =   1  1 − x (T g) ≤ − x (2id−2) = (1−x)2 for .x ∈ [s, 1]. Hence, . T g ≤ (1−id)2 = 31 .

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(f) Using (a), the asserted inequality is verified directly if .α ≤ 2γ = β, and we may −α assume .α < 2γ < β. We conclude that .ξ := 2γ β−α ∈ (0, 1) and define .b : [0, 1] → R by setting  2 +2α(γ −β) 2 b0 (x) := 1 − γβ x + β 2(2γ −α)γ x for x ∈ [0, ξ ], .b(x) := α 2 b1 (x) := 2γ (x − 1) for x ∈ [ξ, 1]. ˘ ` with derivative .b : [0, 1] → R given by It is straightforward to verify that .b ∈ D  2 +2α(γ −β) b0 (x) = − γβ + β (2γ  −α)γ x for x ∈ [0, ξ ], .b (x) = b1 (x) = γα (x − 1) for x ∈ [ξ, 1], which is concave and consists of two lines meeting in the point .(ξ, γα (ξ − 1)). Using (b) and (d), we infer that .s

:= inf{x ∈ (0, 1] : (T g) (x) ≤ b (x)} ∈ [0, ξ ],

x ≥ b |[0,s] and .(T g) |[s,1] ≤ b |[s,1] . Thus .(T g)(x) = 1 + 0 (T g) ≥   1 1 1 + 0 b = b(x) for .x ∈ [0, s] and also .(T g)(x) = − x (T g) ≥ − x b = b(x) for .x ∈ [s, 1], hence 

.(T g) |[0,s]

x

 .

 Tg ≥

1



ξ

b=

0



1

b0 +

0

b1 =

ξ

αβ − 4αγ + 4γ 2 6(β − α)γ

after a tedious but straightforward calculation. Corollary 2.6 . ⊆

√ [ 3−6 3 , 31 ]

  ⊂ (0.2113248,

1 3 ].

˘ ` and .α := − 1 = Proof Let .γ ∈  and let .g ∈ D  solve (Dγ ). Then .g = T g ∈ D g (0)  1 γ := g by Theorem 2.3 and .β := inf g −1 {0} = 1, hence .γ ≤ 3 and .1 ≤ 6(1 − γ )γ by Lemma 2.5(e) and (f), implying .γ ≥

√ 3− 3 6

≈ 0.2113248654.

 

Let us now consider the set .K

:= {g ∈ C˘ : g ≥ 1 − 5 · id,



g ≥ 15 }

and equip it with the metric .d∞ given by .d∞ (f, g)

:= sup |f (x) − g(x)|, x∈[0,1]

that is inherited from .C 0 [0, 1], the space of continuous functions on the interval . Aided by Lemma 2.5, we can establish a few facts about K and its interplay with the operator T .

.[0, 1]

Solving the Iterative Differential Equation .−γ g  = g −1

231

Proposition 2.7 The metric space .(K, d∞ ) has the following properties. (a) K is complete, i.e. closed in .C 0 [0, 1]. (b) .T (K) ⊆ K. (c) The restriction .T |K : K → K is continuous. Proof (a) If .(gn )n∈N is a Cauchy sequence in K, then it converges to a function .g ∈ C 0 [0, 1] Clearly, g is again satisfying .g(0) = 1 and .g(1) = 0 because .C 0 [0, 1] is complete.  decreasing and convex, and both inequalities .g ≥ 1−5·id and . g ≥ 15 hold. Therefore .g ∈ K.  (b) Let .g ∈ K. Then .g ≥ 1 − 5id, .γ := g ≥ 15 and .β := inf g −1 {0} ∈ [ 25 , 1] by Lemma 2.5(a). With Lemma 2.5(f) we infer that . T g ≥ u(γ ), where the function .u : (0, ∞) → R satisfies .u(x)

=

β − 4x + 20x 2 6(5β − 1)x

and u (x) =

20x 2 − β 6(5β − 1)x 2

for all x > 0.

√ √   β  5−1 ˘`⊆ = 23 · 5β−1 ≥ 5−1 > 15 . Moreover, .T g ∈ D We conclude that . T g ≥ u √ 6 2 5 β C˘ and .−(T g) (0) = γ ≤ γ1 ≤ 5 by Lemma 2.5(c), thus .T g ≥ 1 − 5id. In total we have shown .T g ∈ K.  (c) Let .(gn)n∈N be a sequence in K converging to .g ∈ K. Setting .gˆ := g · T g and .g ˆ n := gn · T gn , Remark 2.1(g) yields .|g(x) ˆ

 1   − gˆ n (x)| = x (g ∗ − gn∗ ) ≤ |g ∗ − gn∗ | = |g − gn | ≤ d∞ (g, gn )

ˆ gˆ n ) = 0 and .limn→∞ for all .x ∈ [0, 1], implying .limn→∞ d∞ (g, conclude that .d∞ (T g, T gn ) → 0 as .n → ∞.



gn =



g. We  

3 Intersection Behaviour and Main Results The crucial proofs of Lemma 2.5(e) and (f) rest on the fact that .(T g) intersects another derivative at most once. More generally, if .f, g ∈ D , .g ≤ f = g and ∗ f∗   .(T g) − (Tf ) =  − g g changes its sign only once (from .− to .+ in this case), f then we will have .T g ≤ Tf . To propagate this reasoning to the next iteration step, we would require the difference of .(Tf )∗ and .(T g)∗ , after somehow stretching them vertically, to also change sign at most once. But a vertical stretching of, say .(T g)∗ , ∗ corresponds to a horizontal stretching of T g and thus of .(T g) = − g g , which again corresponds to a vertical and horizontal stretching of g . Because it is hard to tell the stretching factors in advance, we will consider the difference of f and g after arbitrary horizontal and vertical stretching.

232

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As a first step, we want to count how often a given continuous function [a, b] → R defined on a bounded, closed interval .[a, b] changes sign. To this end, we call a closed subinterval .[c, d] ⊆ [a, b] with .a < c ≤ d < b and image . ([c, d]) = {0} a sign switch of . if there exists .δ ∈ (0, min{c − a, b − d}] such that . (c − x) · (d + x) < 0 for all .x ∈ (0, δ]. By .X we denote the set of all sign switches of . and by .χ := #X ∈ N0 ∪ {∞} their number. . :

Remark 3.1 Let .k ∈ N0 , .a, b, a  , b ∈ R with .a < b and .a  < b . Let .u : [a  , b ] → [a, b] and . : [a, b] → R be continuous functions, u bijective. The following statements hold. (a) .X (c ) = X for every .c ∈ R \ {0}. (b) .χ( ◦ u) = χ . (c) .χ ≥ k if and only if there exist .a < x0 < · · · < xk < b such that . (xi−1 ) · (xi ) < 0 for .i ∈ {1, . . . , k}. (d) If . (a) · (b) > 0, then .χ is even or .∞. (e) If . (a) · (b) < 0, then .χ is odd or .∞. (f) Suppose . is continuously differentiable. Then .χ  ≥ χ − 1. If . (a) ·  (a) > 0 in addition, then .χ  ≥ χ . Proof (a)–(c) are immediate from the definition of sign switches. (d) and (e) follow from (c). (f) Suppose that .k := χ ∈ N0 . Then there are .a < x0 < · · · < xk < b as in (c). By the mean value theorem, we can find .yi ∈ [xi−1 , xi ] with .  (yi ) · (xi ) > 0 for  .i ∈ {1, . . . , k}. This shows .χ ≥ k − 1 according to (c).  If . (a) · (a) > 0, then we can find .y0 ∈ [a, x0 ] with .  (y0 ) · (x0 ) > 0; hence  .χ ≥ k by (c) again.  

Given two functions .f, g ∈ D and .a, b > 0, by slight abuse of notation, we introduce the continuous function .f

· a − b · g : [0, min{1, a1 }] → R,

x → f (ax) − bg(x),

the difference between f stretched horizontally by . a1 and g stretched vertically by b. The next lemma tells us how its number of sign switches behaves under swapping f with g and under the operators .∗ and I . Lemma 3.2 For .a, b > 0 and any two functions .f, g ∈ D the following statements hold. (a) .χ(f · a − b · g) = χ(g · a1 − b1 · f ). (b) .χ(f · a − b · g) = χ(g ∗ · b1 − a1 · f∗ ). ˆ := If · a − b · Ig, .b := b ·  f and . := f · a − b · g. Then .χ

ˆ ≤ 1 + χ . (c) Let .

a g ˆ ≤ χ . If .b < 1 < b or .b < 1 < b, then .χ

Solving the Iterative Differential Equation .−γ g  = g −1

233

Proof ˜ := g · (a) Let .a  := min{1, a1 }, . := f · a − b · g and .

˜ .b (ax)

1 a

−

1 b

· f . Then

  1 = b · g( ax a ) − b f (ax) = bg(x) − f (ax) = − (x)

˜ by Remark 3.1(a) and (b). for all .x ∈ [0, a  ]. Hence, .χ = χ

(b) Let .a  := min{1, a1 }, .b := min{1, b}, . := f · a − b · g : [0, a  ] → R and ˜ .

:= g ∗ ·

1 b

−

1 a

· f ∗ : [0, b ] → R.

Because the function .u : [0, a  ] → [0, b ], .x → min{f (ax), bg(x)} is bijective by ˜ ◦ u) = X , the assertion follows from Remark 3.1(b). Remark 2.1(c) and .X (

ˆ is differentiable with continuous derivative (c) According to its definition, .

ˆ .



= a(If ) · a − b · (Ig) = − af ,

ˆ  ≥ χ

ˆ − 1 by Remark 3.1(a) and (f). If .b < 1 < b or .b < 1 < b, hence .χ = χ

 ˆ ˆ (0) = (1 − b) · a · (b − 1) > 0, hence .χ = χ

ˆ  ≥ χ , ˆ again by ·

then . (0) f

Remark 3.1(a) and (f).  

Given .f, g ∈ D and positive ranges .A, B ⊆ (0, ∞), we define the A,B-crossing number B

.χA (f, g)

:= sup{χ(f · a − b · g) : a ∈ A, b ∈ B} ∈ N0 ∪ {∞}

of the ordered pair .(f, g). The ranges .(0, 1), .(0, 1], .[1, ∞) and .(1, ∞), when they appear as lower or upper index to .χ , will be expressed by the symbols ., respectively. Remark 3.3 For .f, g ∈ D and .A, B ⊆ (0, ∞) the following statements hold. (a) (b) (c) (d) (e) (f) (g) (h)

= χA≤ (f, g) and .χA> (f, g) = χA≥ (f, g). = χ≤B (f, g) and .χ>B (f, g) = χ≥B (f, g). and .χ>> (f, g) are even or .∞. and .χ>< (f, g) are odd or .∞. = χ>> (g, f ), and .χ (f, g) = χ>< (g, f ). = χ>> (g ∗ , f ∗ ), and .χ (f, g) = χ (g ∗ , f ∗ ). < .g ≤ f ⇐⇒ χ< (f, g) = 0. > .g ≤ f = g ⇒ χ> (f, g) ≥ 2. < A (f, g) B .χ< (f, g) < .χ< (f, g) > .χ< (f, g) < .χ< (f, g) < .χ< (f, g) .χ

Proof (a) and (b) hold because .f (ax) − bg(x) depends continuously on .a, b ∈ (0, ∞). (c) follows from Remark 3.1(d). (d) follows from Remark 3.1(e). (e) follows from Lemma 3.2(a). (f) follows from Lemma 3.2(b).

234

R. Miyamoto and J. Sander

(g) holds because .g ≤ f implies .f · a − b · g ≥ 0 for all .a, b ∈ (0, 1]. (h) Suppose that .g ≤ f = g and choose .x ∈ [0, 1] with .f (x) − g(x) > 0. Then . := f · a − b · g still satisfies . (x) > 0 for sufficiently small .a, b > 1. Because 1 1 . (0) = 1 − b < 0 and . ( ) = −bg( ) < 0, the assertion follows by Remark 3.1(c). a a  

We are now ready to pin down a relation between two functions in .D , that will be propagated by the operator T . If .f, g ∈ D satisfy .g

≤ f = g,

χ>< (f, g) = 1 = χ (f, g)

χ>> (f, g) = 2,

and

then we write .g  f or .f  g and say that f dominates g . Note that .f

∗

 g ∗ ⇐⇒ f  g ⇐⇒ g ≤ f = g and χ(0,∞) (f, g) ≤ 2 (0,∞)

(3)

by Remark 3.3(a)–(h). Theorem 3.4 Let .f, g ∈ D such that .f  g. Then .If  Ig and .Tf  T g. Proof From .f  g we can conclude .If = Ig because, by Proposition 2.2(c), the operator  f I is injective on .D . Let .a, b > 0, set .k := χ(If · a − b · Ig), .b := ab ·  g and .k



:= χ(f · a − b · g). We distinguish four cases, each time using Lemma 3.2(c).

• Let us first look at the case .a, b < 1. If .b ≤ 1, then .k ≤ 1 + k  ≤ 1 + χ 1, then .k ≤ k  ≤ χ (f, g) = 1 again. Together with Remark 3.3(c) and (g) this proves .χ 1 because . f ≥ g, hence .k ≤ 1 + k  ≤ 1 + χ (f, g) = 2. Together with Remark 3.3(d) this proves .χ (If, Ig) = 1. • Next we look at the case .b < 1 < a. If .b ≤ 1, then .k ≤ 1 + k  = 1 + χ>< (f, g) = 2, and if .b > 1, then .k ≤ k  ≤ χ>> (f, g) = 2 again. Together with Remark 3.3(d) this proves .χ>< (If, Ig) = 1. • Finally, let us consider the case .1 < a, b. If .b < 1, then .k ≤ k  ≤ χ>< (f, g) = 1, and if .b ≥ 1, then .k ≤ 1 + k  ≤ 1 + χ>> (f, g) = 3. Together with Remark 3.3(c) and (h) this proves .χ>> (If, Ig) = 2. Altogether we have shown that If dominates Ig, and .Tf  T g follows by (3).

 

Let us now consider the special sequence of functions .hn

:= T n−1 (1 − id)

for n ∈ N,

(4)

in particular, .h1 (x) = 1 − x , .h2 (x) = (T h1 )(x) = (1 − x)2 and .h3 (x) = (T h2 )(x) = 3 1 − 3x + 2x 2 for .x ∈ [0, 1]. We apply Theorem 3.4 to .h1 and .h2 to show that this sequence descends to a limit function h solving our IDE. Theorem 3.5 The sequence .(hn )n∈N from (4) satisfies .hn+1 ≤ hn for all .n ∈ N and converges uniformly to a function .h ∈ K. Moreover, .h = T h solves (D.κ ) where .κ := h.

Solving the Iterative Differential Equation .−γ g  = g −1

235

Proof We easily verify that .h1 dominates .h2 . With Theorem 3.4, it follows that .hn dominates .hn+1 , hence .hn+1

≤ hn ∈ K

for all n ∈ N

(5)

by Proposition 2.7(b). We conclude that .−5 ≤ hn (0) ≤ hn (x) ≤ 0 for all .x ∈ [0, 1], .n ∈ N. Therefore the sequence .(hn )n∈N is uniformly equicontinuous, and as it is also uniformly bounded, the Arzelà-Ascoli theorem implies that it has a uniformly convergent subsequence. With (5) and Proposition 2.7(a), we conclude that .h := limn→∞ hn ∈ K. Finally, using Proposition 2.7(c), (4) and Theorem 2.3, we see that .T h

solves (D.κ ) with .κ =



=T



 lim hn = lim T hn = lim hn+1 = h

n→∞

n→∞

n→∞

 

h.

Let .h := T ∞ (1 − id) := limn→∞ hn be the solution constructed in Theorem 3.5 and set .κ

:=



h.

˘ ` be a solution to (Dγ ). Then .g ≤ h and .γ ≤ κ. Corollary 3.6 Let .γ ∈  and .g ∈ D Proof From .g  1 − id, we conclude that .g = T n−1 g ≤ T n−1 (1 − id) = hn for  all .n∈ N0 by Theorem 3.4. Using Theorems 3.5 and 2.3, this implies .g ≤ h, thus .γ = g ≤ h =   κ. √

Corollary 3.7 . ⊆ [ 3−6 3 , κ]. Proof This follows from Corollaries 3.6 and 2.6.

 

With a Python program that implements the formula of Proposition 2.2(b) using traverses between .226 + 1 equidistant points and trapezoid integration, we have calculated the approximation .κ

≈ 0.27887706137

by 40 iterations with a numerical precision of 128 bits (approximately 38 decimal places). More precise calculations of this constant will be presented in a follow-up paper. Acknowledgments We want to thank Jürgen Voigt for his sedulous support and his unfailing guidance concerning our numerous (functional) analysis questions. At an early stage in this research, he contributed an elegant, but non-constructive proof using the Schauder fixed-point theorem, that the IDE (Dγ ) has a solution, which eventually we have not included in favour of the more constructive Theorem 3.5.

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References 1. V. Berinde, Existence and approximation of solutions of some first order iterative differential equations. Miskolc Math. Notes 11(1), 13–26 (2010) 2. A. Buic˘a, Existence and continuous dependence of solutions of some functional-differential equations. Semin. Fixed Point Theory Cluj-Napoca 3, 1–14 (1995) 3. E. Eder, The functional-differential equation x  (t) = x(x(t)). J. Differ. Equ. 54(3), 390–400 (1984) 4. E. Egri, I.A. Rus, First order iterative functional-differential equation with parameter. Stud. Univ. Babe¸s-Bolyai Math. 52(4), 67–80 (2007) 5. M. Feˇckan, On a certain type of functional-differential equations. Math. Slovaca 43(1), 39–43 (1993) 6. C. Feng, H. Wang, X.M. Tu, J. Kowalski, A note on generalized inverses of distribution function and quantile transformation. Appl. Math. 3, 2098–2100 (2012) 7. E.P. Klement, R. Mesiar, E. Pap, Quasi- and pseudo-inverses of monotone functions, and the construction of t-norms. Fuzzy Sets Syst. 104(1), 3–13 (1999) 8. L. Levine, Levine’s sequence. https://oeis.org/A011784 (visited on 06 Dec 2022) 9. C.L. Mallows, Triangle of numbers arising from analysis of Levine’s sequence A011784. https://oeis.org/A014621 and https://oeis.org/A144006 (visited on 06 Dec 2022) 10. N.J.A. Sloane, My favorite integer sequences, in Sequences and Their Applications. Discrete Mathematics and Theoretical Computer Science, ed. by C. Ding, T. Helleseth, H. Niederreiter (Springer, London, 1999), p. 118 11. N.J.A. Sloane, C.L. Mallows, B. Poonen, Discussion of A011784. [scans of pages 150–155 and 164 of Sloane’s notebook “Lattices 77” from June–July 1997] https://oeis.org/A011784/ a011784.pdf (visited on 06 Dec 2022)

Irrationality of Zeros of the Digamma Function M. Ram Murty

Dedicated to the memory of Professor Eduard Wirsing

Abstract We prove that all the zeros of the digamma function with at most one possible exception are irrational. Keywords Digamma function · Irrationality

1 Introduction The digamma function .ψ(z) is the logarithmic derivative of the .-function .(z). Thus, ψ(z) =

.

 ∞    (z) 1  1 1 = −γ − − − , (z) z n+z n

(1)

n=1

the second equality arises from the logarithmic differentiation of the Hadamard factorization of the .-function. Here, .γ denotes Euler’s constant. Note that .ψ(z) has poles at the non-positive integers .z = 0, −1, −2, ... and when .z = 1, the series on the right hand side of (1) telescopes and we deduce .ψ(1) = −γ . More generally,

Research of the author is partially supported by an NSERC Discovery grant. M. Ram Murty () Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Maier et al. (eds.), Number Theory in Memory of Eduard Wirsing, https://doi.org/10.1007/978-3-031-31617-3_15

237

238

M. Ram Murty

if m is a natural number, the right hand side of (1) again telescopes and ψ(m) = −γ +

m−1 

.

j =1

1 . j

In an earlier paper, the author and Saradha [5] studied transcendental values of the digamma function. In particular, we showed that for .q > 1, the values ψ(a/q) + γ ,

with

.

(a, q) = 1,

1 ≤ a < q,

(2)

are all transcendental. More precisely, we showed that these numbers are all non-vanishing .Q-linear forms of logarithms of algebraic numbers and hence transcendental by Baker’s theory [1]. This paper can be seen as extending the results of [5] and [6] in another direction. These results are related to a celebrated theorem of Baker, Birch and Wirsing [2]. Chowla asked the question whether there exists a rational valued function .f (n), periodic with prime period p, such that ∞  f (n) .

n=1

n

= 0.

Baker, Birch and Wirsing [2] showed that such a function must be identically zero. In fact, they proved a stronger theorem: if f is a non-vanishing function defined on the integers with algebraic values and period q (not necessarily prime) such that (a) .f (r) = 0 for r satisfying .1 < (r, q) < q; (b) the q q-th cyclotomic polynomial is irreducible over the field .Q(f (1), ..., f (q)), (c) . r=1 f (r) = 0, then ∞  f (n) .

n=1

n

= 0.

In [5], we showed that under these conditions, ∞  f (n) .

n=1

n

=−

q 1 f (a)ψ(a/q), q a=1

and thus the Baker-Birch-Wirsing theorem is related to the digamma function. In fact, the right hand side is a linear form in logarithms of algebraic numbers and its non-vanishing shows that it is a transcendental number. Following [4], it is convenient to define a Baker period as a .Q-linear form of logarithms of algebraic numbers and a general Baker period as any algebraic

Irrationality of Zeros of the Digamma Function

239

number plus a non-zero Baker period. Baker’s theorem [1] then says that if α1 , ..., αm are non-zero algebraic numbers such that

.

.

log α1 , ..., log αm

are linearly independent over .Q, then 1, log α1 , ..., log αm

.

are linearly independent over .Q. In particular, if .α is an algebraic number and . is a non-zero Baker period, then .α +  is transcendental. This remark can be applied to our study of the digamma function. By virtue of the two functional equations 1 ψ(z + 1) = ψ(z) + , z

.

ψ(1 − z) = ψ(z) + π cot π z,

(3)

the result (2) can be extended to all rational numbers as follows. Theorem 1.1 For an arbitrary rational number x which is not an integer, .ψ(x) + γ is a general Baker period which is transcendental. As a consequence, we will deduce: Theorem 1.2 All the zeros of .ψ(x) + γ are real and irrational except for .x = 1. Several questions arise. What are the zeros of .ψ(z) + γ and what is their arithmetic nature? Are they transcendental? By our theorem, they are certainly not rational. What are the zeros of the digamma function and are they transcendental? In this paper, we address these questions. The first observation to make is that taking imaginary parts of (1), we have for .z = x + iy, with .x, y ∈ R, ∞

(ψ(z)) =

.

 y y + , 2 2 x +y (x + n)2 + y 2 n=1

which vanishes if and only if .y = 0. In other words, all the zeros of .ψ(z) and ψ(z) + γ are real. Since

.

ψ  (x) =

.

∞

 1 1 + 2 x (x + n)2 n=0

is positive, .ψ(x) is a strictly increasing function of x in each of the intervals .In = (−n, −n + 1) for .n = 1, 2, .... For notational convenience, we let .I0 = (0, ∞). Thus, there is a unique real zero .xn in each of the intervals .In for .n = 0, 1, 2, ... In particular, there is a unique positive zero .x0 . This observation is a special case of a

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M. Ram Murty

general theorem of Laguerre (see Theorem 2.8.1 on page 23 of [3]). This theorem states the following. If .f (z) is an entire function, not a constant, which is real for real .z and has only real zeros, and is of order 0 or 1, then the zeros of .f  (z) are also real and are separated by the zeros of .f (z). The result above for the digamma function follows from considering .f (z) = 1/ (z). In this paper, we will prove: Theorem 1.3 All the zeros of .ψ(x) defined as .xn are irrational for .n ≥ 0, with at most one exception. We conjecture that they are all in fact transcendental numbers. But we are unable to prove this using our present state of knowledge. What our proof shows is that all the .xn ’s are irrational if .γ is not a general Baker period.

2 Preliminary Results on the Digamma Function In this section, we will review various results needed in the proofs of Theorems 1.2 and 1.3. The first is a famous formula of Gauss (see page 300 of [5]) discovered in 1813. It gives an explicit formula for .ψ(a/q) as a linear form in logarithms of algebraic numbers alluded to earlier. Proposition 2.1 (Gauss, 1813) For .1 ≤ a < q, with .(a, q) = 1, we have πa π cot +2 .ψ(a/q) + γ = − log 2q − 2 q

 0 0, then .x + n − 1 < 0 and again the same argument applies. This completes the proof.

Irrationality of Zeros of the Digamma Function

243

6 Concluding Remarks It would be of interest to show that .x0 is irrational but this looks quite difficult. The positive zero .x0 = 1.461632... can be written as .x0 = 1 + qa with .(a, q) = 1. Thus, .0.45 < a/q < 0.5. By the first of the two functional equations (3), we have 0 = ψ(1 + a/q) = ψ(a/q) +

.

q . a

On the other hand, by Gauss’s formula, we deduce that .γ is a Baker period: q π πa .γ = − log 2q − cot +2 a 2 q

 0 1 if and only if there is an integer .d > 1 such that d divides m and .d k divides a. Lemma 2.1 The Eckford Cohen totient function .ϕ (k) (m) is a multiplicative arithmetic function. For every positive integer m, ϕ

.

(k)

  1 1− k . (m) = m p k

p|m

If p is a prime and v is a positive integer, then

ϕ (k) pv = pvk − p(v−1)k .

.

Proof For every divisor d of m, the number of positive integers up to .mk that are divisible by .d k is .mk /d k . In particular, if .pi1 pi2 · · · pi is a product of distinct primes that divide m, then the number of positive integers up to .mk that are divisible by k k k .(pi1 pi2 · · · pi ) is .m /(pi1 pi2 · · · pi ) . Let .p1 , . . . , pr be the distinct primes that divide m. Let .1 ≤ a ≤ mk . We have k k .(a, m )k = 1 if and only if a is not divisible by .p for all .i ∈ {1, 2, . . . , r}. The i

250

M. B. Nathanson

inclusion-exclusion principle implies that ϕ (k) (m) = mk −

r  mk

.

pik1

i1 =1

=m

k

r 

1−

i=1

r 

+

i1 ,i2 =1 i1 1. vk vk Therefore, .(a, p )k = 1 implies .(a − s, p )k = 1. (k) We have .ds (pv ) = 1 by (1). By Lemma 2.1, we have vk

(k) v .Ms (p )

=

p  a=1 (a,pvk )k =1

  a − s, pvk = pvk − p(v−1)k k

= ϕ (k) (pv ) = ds(k) (pv )ϕ (k) (pv ).

Generalizations of Menon’s Arithmetic Identity

255

the second by .pk , that is, .(s, pk )k = 1. If In

case, the integer s is not divisible k vk k . bp − s, p > 1 for some .b ∈ Z, then .p divides s, which is absurd. Therefore, k .

  bpk − s, pvk = 1 k

for all integers b. Using the gcd sum function .P (k) (m) from Lemma 3.1, we obtain v

(k) v .Ms (p )

=

p 

  a − s, pvk

k

a=1 (a,pk )k =1 p    a − s, pvk − = vk

k

a=1

v

p 

  a − s, pvk

k

a=1 (a,pk )k >1

(v−1)k pvk  p     vk a − s, p bpk − s, pvk = −

k

a=1

b=1

k

p  p   a, pvk − = 1 vk

(v−1)k

k

a=1

b=1

= P (k) (pv ) − p(v−1)k   = (v + 1)pvk − vp(v−1)k − p(v−1)k   = (v + 1) pvk − p(v−1)k



= ds(k) pv ϕ (k) pv .  

This completes the proof. (k)

The arithmetic functions .Ms (m), .ds (m), and .ϕ (k) (m) are multiplicative. Therefore,  

(k) .Ms (m) = Ms(k) pv = ds(k) (pv )ϕ (k) (pv ) pv m

pv m

= ds(k) (m)ϕ (k) (m). This completes the proof of Theorem 1.4. Acknowledgments I thank László Tóth and Kevin O’Bryant for helpful remarks and references.

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M. B. Nathanson

References 1. E. Cohen, Some totient functions. Duke Math. J. 23, 515–522 (1956) 2. P. Haukkanen, Menon’s identity with respect to a generalized divisibility relation. Aequationes Math. 70, 240–246 (2005) 3. P. Haukkanen, J. Wang, A generalization of Menon’s identity with respect to a set of polynomials. Portugal. Math. 53, 331–337 (1996) 4. H.G. Kopetzky, Ein asymptotischer Ausdruck für eine zahlentheoretische Funktion. Monatsh. Math. 84, 213–217 (1977) 5. P.K. Menon, On the sum (a − 1, n), [(a, n) = 1]. J. Indian Math. Soc. (N.S.) 29, 155–163 (1965) 6. K.N. Rao, On certain arithmetical sums, in The Theory of Arithmetic Functions (Proceedings of Conference), Western Michigan University, Kalamazoo, MI, 1971. Springer Lecture Notes in Mathematics, vol. 251 (1972), pp. 181–192 7. I.M. Richards, A remark on the number of cyclic subgroups of a finite group. Am. Math. Monthly 91, 571–572 (1984) 8. L. Tóth, Proofs, generalizations and analogs of Menon’s identity: a survey (2021). arXiv:2110.07271

On a Conjecture of Descartes János Pintz

To the memory of Eduard Wirsing

Abstract Descartes conjectured a century before Goldbach a similar, but different conjecture. According to this, every even number can be written as the sum of at most three pimes. It is easy to see that this is equivalent with the conjecture that for every even N at least one of N and .N + 2 is the sum of two primes. At present the conjecture seems to be hopeless. (The best result in this direction is that at least one of .N, N +2, N +4...N +M can be written as the sum of two primes where .M = N b for a b approximately 1/20.) This implies that the size .D(X) of the exceptional set for Descartes conjecture (for even numbers below a large number X) is at most of the size .E(X) of the exceptional set for the Goldbach conjecture. Earlier methods were unable to estimate .D(X) better than .E(X). We prove that for any .c > 3/5 we have .D(X) = O(Xc ) which is stronger than the best results for .E(X). Keywords Descartes conjecture · Goldbach conjecture · Exceptional set in Goldbach’s problem

1 Introduction It is well known that during the correspondence of Euler and Goldbach the following conjecture—today known as Goldbach’s conjecture (or sometimes called even Goldbach conjecture)—was formulated in 1742.

Supported by National Research Development and Innovation Office, NKFIH, K 119528 and KKP 133819. J. Pintz () ELKH Rényi Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Hungary e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Maier et al. (eds.), Number Theory in Memory of Eduard Wirsing, https://doi.org/10.1007/978-3-031-31617-3_17

257

258

J. Pintz

Goldbach Conjecture (Binary Goldbach Conjecture) Every even number greater than 2 can be written as the sum of two primes. In his original letter Goldbach formulated two similar, more complicated conjectures which were actually equivalent with the above and it was Euler who used the above formulation in his reply letter. However, he noted in the same letter that Goldbach mentioned him earlier in a conversation the above simpler and more elegant form. So it is fully justified to attribute the conjecture to Goldbach. It is much less known—I learned it from a manuscript of D. Wolke—that Descartes (1591–1650) mentioned many years before the following assertion (without any proof), what we will call Descartes Conjecture. Descartes Conjecture Every even number can be expressed as the sum of at most three primes. This assertion appeared in print first in the collected works of Descartes only in the 1908 edition ([3], Opuscula Posthuma, Excerpta Mathematica, Vol. 10, p. 298), so we can rightly assume that Goldbach and Euler did not hear about this before their correspondence in 1742. Descartes does not mention odd numbers in his assertion, but the same assertion follows trivially for odd numbers from the assertion for even numbers. It is also obvious that if an even N satisfies Descartes Conjecture then N or .N −2 can be expressed as the sum of two primes. The converse is clearly also true. In the present work we will investigate the number of possible exceptional Descartes numbers below a large bound X, that is (.P denotes the set of primes)   j  .D(X) = # n ≤ X; 2 | n, n = pi , pi ∈ P , for j ≤ 3 .

(1)

i=1

It is trivial that .D(X) ≤ E(X), where E(X) = # {n ≤ X; 2 | n, n = p1 + p2 , pi ∈ P }

.

(2)

is the size of the exceptional set for Goldbach’s problem. The strongest published result E(X)  X0.879

.

(3)

is due to Wen Chao Lu [8]. We improved this to E(X)  X0.72

.

in a work in arXiv [11].

(4)

On a Conjecture of Descartes

259

Our present goal is to show a sharper estimate for .D(X). Earlier methods did not allow to prove a distinctly sharper bound for even exceptional Goldbach numbers n if we knew that .n − 2 is also an exceptional Goldbach number. The crucial point, which makes a more effective treatment of .D(X) possible is an approximate formula for the contribution of the major arcs [10]. This formula shows that a particular .L -zero close to the line .Re s = 1 can have a bad effect for the number of Goldbach decomposition of an even number n (more precisely, for the contribution of the major arcs to it), but not simultaneously for n and .n − 2. We will prove Theorem 1.1 .D(X) ε X3/5+ε for any .ε > 0.

2 Notation: The Role of the Explicit Formula The explicit formula proved in [10] will play a central role in the proof of Theorem 1.1; in order to formulate it we first need to introduce the notation. Let .ε and .ε0 be small positive numbers, X be a number large enough .(X > X0 (ε, ε0 )), and let us define X1 := X 1−ε0 , e(u) := e2π iu , S(α) :=



.

log p e(pα), L = log X,

(5)

X1 X (ε) .2 | m ∈ 0 2 , X . Then there exists .P ∈ (X R1 (m) =

 

.

A(i )A(j )S(χi , χj , m)

i ∈E j ∈E



+ Oε Xe

−cH

(i ) (j ) i +j −1 m (i + j )

(14)

 X 1−ε , + √ +X T

where the generalized singular series satisfy |S(χi , χj , m)| ≤ S(χ0 , χ0 , m) = S(m);

.

(15)

further for any .η small enough |S(χi , χj , m)| ≤ η,

.

(16)

unless the following three conditions all hold, ri |C(η)m, rj |C(η)m, cond χi χj < η−3

.

(17)

where .C(η) is a suitable constant depending only on .η. Its proof follows from Theorem 1 [10] and Main Lemma 1 of [10]. Remark 1 A very important feature of the explicit formula is that the number K of generalized exceptional zeros appearing in (14) is by the log-free zero density theorem of Jutila [4]. N ∗ (α, T , Q) ε (Q2 T )(2+ε)(1−α) for ε > 0, α ≥ 4/5

(18)

K ≤ Ce2H ,

(19)

.

from which .

so it is bounded by an absolute constant (depending on .ε), if we choose H as a sufficiently large absolute constant depending on .ε, which we suppose later on in the proof of Theorem 1.1. Similarly, we will choose T as a sufficiently large constant depending on .ε. Remark 2 A very important information of the explicit formula is the relation (17) which shows that a generalized exceptional character .χi causes a problem only for the quasi-multiples m of its conductor .ri .

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J. Pintz

Although the quoted explicit formula is in general a good starting point for the proof of (20)

R1 (m) > εS(m)m

.

if .ϑ is small enough, the argument breaks down in case of the existence of a Siegelzero .1 − δ corresponding to .L(s, χ1 ), in which case we might have .S(χ1 , χ1 , m) = −S(m) and we cannot show the crucial relation (26) if .δ is small enough. In this case the Deuring–Heilbronn phenomenon can help. This case was worked out as Theorem 2 in [10] which we quote now as Theorem B Let .ε > 0 be arbitrary. If .X > X(ε ), ineffective constant and there exists a Siegel zero .β1 of .L(s, χ1 ) with 4



β1 > 1 − h/ log X, cond χ1 ≤ X 9 −ε ,

.

(21)

where h is a sufficiently small constant depending on .ε , then 3



E(X) < X 5 +ε .

.

(22)

Remark 3 Let us fix a sufficiently small .ε > 0. Then in the proof of Theorem 1 we are entitled to suppose that all .L(s, χ ) functions . mod r ≤ P satisfy L(s, χ ) = 0

.

for s ∈ [1 − c0 / log X, 1]

(23)

if we choose .ϑ ≤ 0.44. In other words, we can suppose that there are no exceptional zeros .1 − δ satisfying .δ < c0 / log X with a small but fixed .c0 > 0. The well-known relation (cf. [5], p. 46) .(Re w, Re z > 0) (w) (z) = B(w, z) = . (w + z)

1 x w−1 (1 − x)z−1 dx

(24)

0

tells us that |B(i , j )| ≤ |B(Re i , Re j )| = B(1, 1) + O(1/ log X) = 1 + O(1/ log X). (25)

.

Hence, taking into account the relations (15)–(17) we see that the estimation (20) will follow, if we can show ∗ .

i ,j ∈E (i ,j )=(1,1)

5 X−δi −δj ≤ 1 − ε, 2

(26)

On a Conjecture of Descartes

263

where the .∗ means that the additional condition (17) is satisfied for the pairs .(i , j ) of zeros in the summation with .η chosen as in (33) of Section 4. The expression (26) can be estimated directly by density theorems and the Deuring–Heilbronn phenomenon, as done in the earlier estimates of Chen-Liu [1], Hongze Li [6, 7], and Lu [8]. It also resembles the well-studied problem of the Linnik-constant, with the seemingly major disadvantage that the zeros do not belong to a fixed modulus q ≤ P

.

(†)

but to a set of different moduli .ri ≤ P . During the proof we will show that this disadvantage can be overwhelmed thanks to the information (17) supplied by the explicit formula.

3 Contribution of the Minor Arcs We will use the same treatment for the minor arcs as all earlier works beginning with the pioneering one of I. M. Vinogradov [13] in which he proved the ternary Goldbach conjecture, the so-called three primes theorem for every sufficiently large odd numbers, i.e. 2N + 1 = p1 + p2 + p3 , pi ∈ P , for N > N0 .

.

(27)

This is based for his estimation of trigonometric sums for primes, simplified later by Vaughan (see [2], Chapter 25)

S(α) 

.

    1 a  X 4/5 1/2 4  log X if α −  ≤ 2 , (a, q) = 1. √ + X + (Xq) q q q (28)

This implies by Parseval’s identity  .

m≤x

 |S 4 (α)|dα

R22 (m) =

(29)

m  2 ≤ max |S(α)| m



1 |S(α)|2 dα  max

 X2 8/5 XL 9 . ,X P

0

Vinogradov chose .P = L A (with any large A). This choice makes an asymptotic evaluation of .R 1 (n) possible for all .n ≤ X. On the other hand, in this case we get a relatively weak upper estimate for the contribution of the minor arcs due to the moderate size of P (cf. (29)). It was the idea of Vaughan [12] and Montgomery–

264

J. Pintz

Vaughan [9] to choose P larger. However, then we lose the possibility of asymptotic evaluation of .R1 (n) due to the possible existence of a Siegel-zero. The situation is somewhat easier by a result of Landau and Page (see [2], Chapter 14) according to which for a given large X we might have only at most one Siegel-zero with a character with conductor .≤ X. Then the idea of [12] and [9] was to evaluate the effect of the possible single Siegel-zero for .R1 (n). In [9] they are able to choose c .P = X in such a way with a small fixed absolute constant .c > 0. In our present work we are able to work with a .P = Xϑ for any fixed constant .ϑ < 4/9, e.g. with .ϑ = 0.4 or .0.44. Consequently (choosing .P ≥ X2/5 ) we have     R2 (m) ≤ X1−ε with O ε X3/5+3ε exceptions.

.

(30)

4 Proof of Theorem 1.1 We will choose .P0 = Xϑ+2ε , so our P will satisfy   P ∈ Xϑ+ε , Xϑ+2ε .

.

(31)

Thus the exceptional set arising from the minor arcs (10) will be .o(X1−ϑ ) (cf. (29)– (30)). We will distinguish two cases. Case 1 All zeros of all .L -functions with a conductor .≤ P satisfy .δ ≥ 5ε/L i.e. .β = Re  ≤ 1 − 5ε/L . Case 2 There exists a (real) Siegel zero with a conductor .≤ P satisfying .δ < 5ε/L i.e. .β > 1 − 5ε/L . In Case 1 we consider the set .R of the K generalized exceptional zeros appearing in (14) whose number K is bounded by an absolute constant depending on .ε, 0 ≤ K ≤ K(ε) − 1

.

(32)

according to (19) since we will choose H as a big constant depending on .ε. (If K = 0 we are ready.) Let us choose now

.

η=

.

ε , K 2 (ε)

(33)

and write C(η) = C1 (ε).

.

(34)

On a Conjecture of Descartes

265

In this case the total contribution of terms not satisfying (17) will be really less than εX in (14), so (26) will really imply (20). Let us divide now the even numbers m in |R | different classes .M (R  ) according to the subset .R  ⊂ R .[X/2, X] into at most .2 of generalized exceptional zeros which belong to primitive characters with moduli dividing .C1 (ε)m .

  M (R  ) = m ∈ [X/2, X], 2 | m, ri | C1 (ε)m ⇔ ri ∈ R  .

.

(35)

(The subset might be empty for some .R  ⊂ R ; for example, if .l.c.m.[ri ] > XC1 (ε).) ri ∈R 

We have clearly  q(R  ) := l.c.m.[ri ; ri ∈ R  ]  C1 (ε)m for m ∈ M (R  ).

.

(36)

Let us consider now a pair of classes .R 1 , R 2 and the quantities M (R 1 ), M (R2 ), q(R 1 ), qR 2 ).

.

(37)

From (35)–(37), applied for m and .m − 2 we obtain with the notation   g.c.d. q(R 1 ), q(R 2 ) = d, q(R 1 ) = q1 d, q(R 2 ) = q2 d

.

(38)

the relation d | 2C1 (ε).

.

(39)

Hence C1 (ε)m ≡ 0 (mod q1 ),

.

C1 (ε)m ≡ 2C1 (ε) (mod q2 )

(40)

which implies that there is an .aε (m) with C1 (ε)m ≡ aε (m)

.

(mod q1 q2 ).

(41)

The number of .m ≤ x with (41) is by (38)–(41) .

ε

X + 1. q(R 1 )q(R 2 )

(42)

This means that from the point of proving Theorem 1.1 we can restrict our attention to the case when .

  min q(R 1 ), q(R 2 ) ≤ X1/5 .

(43)

266

J. Pintz

Summarizing the content of Sects. 2–4 let us suppose that

P ∈ X2/5 , X2/5+ε ,

.

m≤X

and R(m) = R(m − 2) = 0.

(44)

.

Taking into account that the mean square of the contribution .R2 (m) of the minor arcs is small in case of .P ≥ X2/5 , i.e., by (29)–(30) it is sufficient to show that .

  max R1 (m), R1 (m − 2) > XL −1/2 .

(45)

The arguments of (37)–(43) show that apart from a possible exceptional set of size .O (X3/5+ε ) we can suppose that (43) holds, so WLOG we can assume that q0 := q(R 1 ) ≤ X1/5 ,

.

A := log X/ log q0 ≥ 5.

(46)

However, in Case 1, i.e. if there are no Siegel-zeros then the main result of [11] asserts that if (46) holds, then by restricting the set .E to .E  for .L -zeros with .cond χ | q0 and .cond χi χj < C0 (ε) we have to show 

S :=

.

i ,j ∈E  ,(i ,j )=(1,1)

−A(δi +δj )

q0

< 1 − 5ε.

(47)

This is proved in [11] even for .A = 25 7 . The case when the LHS of (47) is maximal is treated in (9.36)–(9.37) of [11] (cf. the notation (2.38) of [11]) and yields .(λ1 = δ1 log q0 ) for .A = 5, .λ1 small S ≤ e−2Aλ1 + 5λ1 < 1 − Aλ1 /2 = 1 − δ1 log X/2 ≤ 1 − 5ε/2.

.

(48)

The arguments of Sect. 2, actually a summary of the results of [10], show that (47) really proves that if m is not in an exceptional set of size .O(X3/5+ε ) estimated in (42) then R1 (m) > (1 − ε)mS(m)

.

(49)



Repeating the arguments for the intervals . X · 2−ν−1 , X · 2−ν we obtain D(X) ε X3/5+ε

.

(50)

Finally in Case 2 Theorem B proves (50) even in the sharper form .E(X) ε X3/5+ε which clearly implies the same inequality for .E(X) replaced by .D(X), i.e. (50).

On a Conjecture of Descartes

267

References 1. J.R. Chen, J.M. Liu, The exceptional set of Goldbach numbers III. Chinese Quart. J. Math. 4, 1–15 (1989) 2. H. Davenport, Multiplicative Number Theory, 2nd edn. Revised by H.L. Montgomery, Graduate Texts in Mathematics, vol. 74 (Springer, New York, 1980), xiii+177 pp. 3. R. Descartes, Oeuvres (Publiées par Ch. Adam et P. Tannery, Paris, 1908) 4. M. Jutila, On Linnik’s constant. Math. Scand. 41, 45–62 (1975) 5. A.A. Karatsuba, Basic Analytic Number Theory. Translated from the second (1983) Russian edition and with a preface by Melvyn B. Nathanson (Springer, Berlin, 1993), xiv+222 pp. 6. H. Li, The exceptional set of Goldbach numbers I. Quart J. Math. Oxford Ser. (2) 50(200), 471–482 (2000) 7. H. Li, The exceptional set of Goldbach numbers II. Acta Arith. 92(1), 71–88 (2000) 8. W.C. Lu, Exceptional set of Goldbach number. J. Number Theory 130(10), 2359–2392 (2010) 9. H.L. Montgomery, R.C. Vaughan, The exceptional set in Goldbach’s problem. Collection of articles in memory of Juri˘i Vladimiroviˇc Linnik, Acta Arith. 27, 353–370 (1975) 10. J. Pintz, A new explicit formula in the additive theory of primes with applications I. The explicit formula for the Goldbach and Generalized Twin Prime Problems. Acta Arith. (to appear). arXiv: 1804.05561 11. J. Pintz, A new explicit formula in the additive theory of primes with applications II. The exceptional set in Goldbach’s problem (2018). arXiv: 1804.09084 12. R.C. Vaughan, On Goldbach’s problem. Acta Arith. 22, 21–48 (1972) 13. I.M. Vinogradov, Representation of an odd number as a sum of three prime numbers. Doklady Akad. Nauk SSSR 15, 291–294 (1937; Russian)

On the Greatest Common Divisor of a Number and Its Sum of Divisors, II Paul Pollack

In memory of Eduard Wirsing, with appreciation and admiration

Abstract Let .E(x, y) = #{n ≤ x : gcd(n, σ (n)) > y}. We collect known results about the distribution of .E(x, y) and establish a new, sharp estimate for .E(x, y) when y grows faster than any power of .log log x but .y = exp((log log x)o(1) ). Taken together, these results determine the order of magnitude of .log(E(x, y)/x) whenever .1 ≤ y ≤ x 1− . Keywords Perfect number · Multiperfect number · Multiply perfect · Sum of divisors

1 Introduction 1.1 Perfect Numbers A natural number n is called perfect if .σ (n) = 2n; equivalently, n is perfect if n is the sum of its proper divisors. Perfect numbers appear already in Euclid’s Elements (ca. 300 BCE), where it is shown that .2k−1 (2k − 1) is perfect whenever .2k − 1 is prime. Two thousand years later, Euler established a partial converse to Euclid’s theorem: Every even perfect number is given by Euclid’s formula. To this day, no odd perfect numbers are known, and deciding whether any exist stands as perhaps the oldest unsolved problem in number theory. More modestly, one might hope to show that if odd perfect numbers exist, at least there cannot be too many of them. To quantify this, let .V (x) denote the number of perfect numbers .n ≤ x. (While even perfect numbers are counted in .V (x), the count of

P. Pollack () Department of Mathematics, University of Georgia, Athens, GA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Maier et al. (eds.), Number Theory in Memory of Eduard Wirsing, https://doi.org/10.1007/978-3-031-31617-3_18

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even perfect numbers in .[1, x] is .O(log x), which is dwarfed by all of the upper bounds on .V (x) to be discussed.) In 1933, Davenport [2] showed that .n/σ (n) has a continuous distribution function. That is, for each .u ∈ [0, 1], the asymptotic density of n with .n/σ (n) ≤ u exists, and this density varies continuously with u. The continuity in Davenport’s result implies that for any fixed real number .μ, the n with 1 .n/σ (n) = μ make up a set of density 0. In particular (.μ = 2 ), .V (x) = o(x), as .x → ∞. In 1954, Kanold gave a more direct proof that .V (x) = o(x) [10]. Kanold’s contemporaries seem to have viewed his paper as throwing down the proverbial gauntlet, prompting a flurry of improved bounds for .V (x) over the next several years, collected in Proposition 1.1. Proposition 1.1 We have the following upper bounds for .V (x). All O-estimates are to be understood as holding when x is sufficiently large. Researcher(s) Volkmann [20] Hornfeck [8] Kanold [11] Erd˝os [6]

Year 1955 1955 1956 1956

Kanold [12] Hornfeck and Wirsing [9]

1957 1957

Estimate = O(x 5/6 ) 1/2 for all .x > 0 .V (x) < x 1/2 ), as .x → ∞ .V (x) = o(x 1/2−δ ), .V (x) = O(x for some .δ > 0 1/4 log x/ log log x) .V (x) = O(x  .V (x) = O (x );   .V (x)

O

1959

Wirsing [21]

log log log x

log log x in fact, .V (x) ≤ x O(1/ log log x) .V (x) ≤ x

Incredibly, Wirsing is still ‘winner and world champion’ as far as upper bounds on .V (x): Despite six decades of further investigations, we still have no proof that (x) for a function .(x) = o(1/ log log x). .V (x) ≤ x It seems appropriate given the nature of this volume to sketch a version of Wirsing’s ingenious argument. Suppose that .n ≤ x is perfect, and suppose also that we have in hand a unitary divisor1 d of n with .d > 1. Then either .σ (d) = 2d, in which case .n = d, or .d < σ (d) < 2d, in which case . σ2d (d) has a lowest-terms denominator larger than 1. Since . σ2d (d) =

σ (n/d) n/d ,

if we choose .p1 as the least prime

2d . σ (d)

(which, it should be noted, depends only on d), dividing the denominator of then .p1 divides .n/d. We let .e1 be the positive integer for which .p1e1  n/d and start the argument over with our unitary divisor d replaced by the new unitary divisor e1 .dp . We continue in the same way until our unitary divisor reaches n itself, at 1 which point we have discovered a factorization n = dp1e1 · · · pkek

.

1 Meaning, .d

| n and .gcd(d, n/d) = 1.

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where each .pi is entirely determined by of d and the exponents .ej for .j < i. Then n itself is determined by d and the exponent sequence .e1 , . . . ek . Wirsing takes d as the .log x-smooth part of n (which, as is not difficult to show, must exceed 1 once x is large), and he derives his .x O(1/ log log x) estimate by bounding above the number of choices for d and for the exponent sequence .e1 , . . . , ek . Wirsing’s results in [21] extend somewhat beyond an upper bound on .V (x). What is actually shown is that every equation .σ (n) = λn has at most .x O(1/ log log x) solutions .n ≤ x, once .x ≥ 3, where the implied constant is independent of .λ. The independence of the bound on .λ can be crucial in applications (such as Lemma 3.2 below; another example is the main result of the recent paper [15]). One easy consequence of this uniformity is that the number of .n ≤ x that are multiply perfect—meaning that .σ (n)/n ∈ Z—is also bounded by .x O(1/ log log x) .

1.2 The Distribution of gcd(n, σ (n)) The just mentioned consequence of Wirsing’s theorem for multiply perfect numbers can be read as saying that there are very few n with .gcd(n, σ (n)) as large as possible. In this note we are interested more generally in the distribution of .gcd(n, σ (n)) as n ranges through the integers in .[1, x]. It was Erd˝os who opened up this line of investigation. In [5], Erd˝os shows that the number of .n ≤ x with .gcd(n, σ (n)) = 1 is .∼ e−γ x/ log log log x, as .x → ∞, where 2 .γ is the Euler–Mascheroni constant. In Sect. 2, we present a souped-up version of Erd˝os’s argument, proving that the count of .n ≤ x with .gcd(n, σ (n)) = m is −γ x/m log log log x, uniformly for .m ≤ (log log x)1/4 . .∼ e Larger values of .gcd(n, σ (n)), but still of size bounded by a power of .log log x, were considered by Erd˝os in [6]. To ease notation, define E(x, y) = #{n ≤ x : gcd(n, σ (n)) > y}.

.

Theorem 4 of [6] asserts the existence of a continuous function .D(u), strictly decreasing on .(0, ∞), such that for each positive real number u, E(x, (log log x)u ) ∼ D(u)x,

.

(1)

as .x → ∞. While the proof is omitted in [6], details appear in later joint work with Luca and Pomerance [7] where it is shown that D(u) = e−γ



∞

ρ(t) dt,

.

(2)

u

2 Erd˝ os

states his result for .gcd(n, ϕ(n)) rather than .gcd(n, σ (n)), but the argument for .σ (n) is very similar. See also [17].

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with .ρ(u) the Dickman function from ‘smooth number’ theory: the continuous solution on .(0, ∞) to the difference delay equation .ρ (u) = −ρ(u−1)/u for .u > 1, with .ρ(u) = 1 for .0 < u ≤ 1. The relation (1) is a consequence of another result of [7], of independent interest, that .gcd(n, σ (n)) is the .log log x-smooth part of n for all but .o(x) values of .n ≤ x.3 Though not stated explicitly there, the argument in [7] establishes that the asymptotic relation (1) holds uniformly in u, for u restricted to any compact subinterval of .(0, ∞). Another claim of [6], again stated without proof (see Theorem 3 there and the subsequent remarks), is that .E(x, y) undergoes a phase transition as y grows beyond o(1) . A corrected version of this claim is established in [14], where it is .y = (log x) shown that the true threshold is .y = exp((log log x)o(1) ). Proposition 1.2 (See Theorems 1.1 and 1.2 in [14]) If .x → ∞ and .y = exp((log log x)o(1) ), then E(x, y) > x/y o(1) ,

.

while for each .β > 0 there is a constant .c = c(β) > 0 with E(x, y) < x/y c

.

whenever y > exp((log log x)β ) and x is large.

As an illustration of the second half of Proposition 1.2, if .y > exp((log log x)1/3 ), one can deduce from the proofs in [14] that .E(x, y) < xy −1/1000 once x is sufficiently large. The shape of the upper bound—x divided by a constant power of y—is best possible, up to the precise constant, since it is shown in [14, Theorem 1.4] that .E(x, y) > x/y 1+o(1) when .x → ∞ and .2 ≤ y ≤ x 1− (for any fixed . > 0). The above results give only weak information about .E(x, y) when y tends to infinity faster than any power of .log log x but .y = exp((log log x)o(1) ). Our main theorem addresses this missing range. Theorem 1.1 If .x → ∞ and .u := then

log y log log log x

→ ∞, with .y ≤ exp((log log x)o(1) ),

E(x, y) = x exp (− (1 + o(1)) u log u) .

.

We also prove a somewhat weaker estimate in the wider range .y exp((log log x)1− ).

3 These

results of [7] are stated for .ϕ(n), but the proofs for .σ (n) are essentially the same.

(3) ≤

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log y log log log x

→ ∞, with .y ≤

Theorem 1.2 Fix . > 0. If .x → ∞ and .u := exp((log log x)1− ), then

x exp (− (1 + o(1)) u log u) ≤ E(x, y) ≤ x exp (− (1/7 + o(1)) u log u) .

.

(4)

We do not know if (3) holds in the entire range of Theorem 1.2. The function .D(u), as defined in (2), can be shown to satisfy .D(u) = exp(−(1 + o(1))u log u) as .u → ∞ (cf. the arguments of Sect. 3). Thus Theorems 1.1 and 1.2 assert that weaker versions of (1) hold in extended ranges of y. For the lower bounds, and for the upper bound when y is small, we prove Theorems 1.1 and 1.2 by borrowing ideas from the proof of (1). For larger values of y, we obtain the upper bounds by adapting the method used by Erd˝os in [6] to bound the counts of perfect and multiperfect numbers. These arguments of Erd˝os were also the basis of much of the work in [14].  In [14], Wirsing’s theorem is used to deduce that . x1 n≤x gcd(n, σ (n)) ≤ √ x O(1/ log log x) . (This result is quoted √as Lemma 3.2 below.) Thus if y tends to infinity faster than any power of .x 1/ log log x , then .E(x, y) < x/y 1+o(1) . That requirement on y is surely too stringent; it would be very interesting to know the true threshold for y after which the savings of .y 1+o(1) ‘kicks in’. Perhaps it suffices for .log y to grow faster than any power of .log log x. Perhaps even 1+ ) is enough? It follows from Theorem 1.2 that .y > .y > exp((log log x) 1− exp((log log x) ) is not sufficient.

1.3 A Word on Notation We write .A  B to mean .A ≥ (1 + o(1))B; naturally, .A  B means .B  A. The letters p and . (but not q) are reserved for primes.

2 The Frequency of n with gcd(n, σ (n)) = m In this section we prove the claim made in the introduction that #{n ≤ x : gcd(n, σ (n)) = m} ∼ e−γ

.

x , m log log log x

(5)

as .x → ∞, uniformly for .m ≤ (log log x)1/4 . We have not seen this result in the literature, but the method of proof follows [5] and [7] closely. The argument is included for completeness, and to clear ground for the proofs of Theorems 1.1 and 1.2 in Sect. 3.

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The following lemma is due to Pomerance (see [16, Theorem 3] and its application on p. 221 there). Lemma 2.1 For all .x ≥ 3 and each positive integer .d ≤ x, the number of .n ≤ x for which .d  σ (n) is .O(x/(log x)1/ϕ(d) ). Here the implied constant is absolute. Put Z :=

.

log log x , 2 log log log x

(6)

and let L = lcm {d : 1 ≤ d ≤ Z}.

.

(7)

Let m be a divisor of L not exceeding .x 1/2 . (This certainly allows all .m ≤ (log log x)1/4 , once x is large. The extra generality will be useful later.) We consider .n ≤ x of the form .n = mq, where every prime dividing q exceeds Z. Performing inclusion-exclusion over the primes dividing Z, one finds that there x  −γ x/m log log log x such n. By Lemma 2.1, the number are .∼ m

≤Z (1−1/ ) ∼ e of these n for which .σ (q) is not divisible by L is .



x  x x  , (log (x/m))−1/ϕ(d)  (log x)−1/Z ≤ m m m log log x d≤Z

d≤Z

which is .o(x/m log log log x). Hence, there are .∼ e−γ x/m log log log x values .n = mq ≤ x with q having all prime divisors greater than Z and with .σ (q) divisible by L. All of these n are such that .m | gcd(n, σ (n)). From now on, we assume that .m ≤ (log log x)1/4 . Let us show that almost all of the n constructed above have .gcd(n, σ (n)) = m. If m is a proper divisor of .gcd(n, σ (n)), then . | gcd(n, σ (n)) for some prime . > Z. Since .Z > σ (m), it must be that this prime . divides .gcd(q, σ (q)). We may assume q is squarefree; otherwise n has squarefull part at least .Z 2 , and the number of such n is .O(x/Z), which is .o(x/m log log log x). Thus there are primes . , p > Z dividing q with .p ≡ −1 (mod ). The number of such .q ≤ x/m for which . > Z := log log x · log log log x is .



x  m



p≤x

>Z p≡−1 (mod )

x  log log x 1 x   , 2 p

m

m(log log log x)2

(8)

>Z

 which is also .o(x/m log log log x). (We used here that . p≤x, p≡−1 (mod ) 1/p  (log log x)/ , which follows from the Brun–Titchmarsh theorem by partial summation.) If instead . ∈ (Z, Z ], write .q = q . Since .q ≤ x/m and .q is free of prime factors up to Z, there are . x/m log log log x possibilities for .q given . . Summing on . ∈ (Z, Z ] shows that there are .o(x/m log log log x) integers .n = mq

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of this kind as well. Collecting our results we arrive at the lower bound implicit in (5). To prove the upper bound, suppose .gcd(n, σ (n)) = m, and write .n = mq. It suffices to show that, with .o(x/m log log log x) exceptions, q is free of prime factors below Z. Write .q = q1 q2 , where every prime dividing .q1 divides m and 2 3 .gcd(q2 , m) = 1. If .q1 > m (log log log x) , then n has a squarefull divisor larger 2 3 than .m (log log log x) , putting n in a set of size .o(x/m log log log x). So we may suppose that .q1 ≤ m2 (log log log x)3 . We may also suppose that .L | σ (q2 ), since the number of exceptional q is .



 q1 d≤Z

x/mq1 x  1  x .  (log x)−1/Z  1/ϕ(d) m q q1 m log log x (log(x/mq1 )) 1

d≤Z

Since .mq1 < Z and .L | σ (q2 ) | σ (n), we see that .mq1 | gcd(n, σ (n)) = m, forcing q1 = 1. Thus, .q = q2 ; that is, q is prime to m. If q has a prime factor .p ≤ Z, then .p  m, and .pm | L (since both .p, m | L). But then .pm | gcd(n, σ (n)), contradicting that .gcd(n, σ (n)) = m. .

3 Proof of Theorem 1.1 3.1 Preliminaries Concerning Smooth Numbers We begin by collecting certain statements from ‘smooth number’ theory that will prove useful momentarily. Let .(X, Y ) denote the count of Y -smooth .n ≤ X, and let .2 (X, Y ) denote the same count restricted to squarefree n. Fix . > 0. It is known that whenever log X 1+ , we have .X, Y → ∞ with .U := log Y → ∞ and .Y ≥ (log X) X exp(−(1+o(1))U log U ) ≤ 2 (X, Y ) ≤ (X, Y ) ≤ X exp(−(1+o(1))U log U ),

.

where .U := log X/ log Y . Here the upper bound follows from [3, Theorem 2] while the lower bound is a consequence of the lower bound on .(X, Y ) in [1, Theorem 3.1] combined with [13, Théorème 2], which estimates the ratio .(X, Y )/2 (X, Y ). See also [4, Théorème 2.1]. Recall also that (X, Y ) ∼ Xρ(U ) if X, Y → ∞ with X ≥ Y ≥ exp((log log X)2 );

.

(9)

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P. Pollack

furthermore, ρ(U ) = exp(−(1 + o(1))U log U ),

as U → ∞,

ρ(U ) , U log U

as U → ∞.

.

ρ(U + 1) ∼

.

and

(10) (11)

For proofs of (9)–(11), see Chapter III.5 of [19]; the relation (11) is proved by combining equations (5.48) and (5.62) there.

3.2 Lower Bounds in Theorems 1.1 and 1.2 log y We shall prove that .E(x, y) ≥ x exp(−(1+o(1))u log u) whenever .u = log log log x → 1− ∞ and .y ≤ exp((log log x) ). To start off, assume that .u ≥ log log log log x; that is, .y ≥ (log log x)log log log log x . Let Z and L be as defined in (6) and (7). Consider n of the form .n = mq, where .m ∈ (y, x 1/2 ] is both squarefree and Z-smooth, and where .q ≤ x/m has all prime factors exceeding Z. By our work in §3, for each m there are −γ x/m log log log x corresponding values of q, and this remains true if we also .∼ e require that L divides .σ (q). Then .m | L | σ (q) | σ (n), so that .gcd(n, σ (n)) ≥ m > y. Moreover, the number of n produced this way is

.

∼ e−γ

x log log log x

 m∈(y,x 1/2 ] m Z-smooth

μ2 (m) . m

Since .exp(u log u) ≥ (log log log x)log log log log log x , the denominator .log log log x has the shape .exp(o(u log u)). Thus, it suffices to bound the sum from below by .exp(−(1 + o(1))u log u). Put .U = log y/ log Z, so that .U  (log log x)1− / log log log x. We fix .δ > 0 and define .y = y exp(δU log U ). Then .y ≤ exp((log log x)1−+o(1) ). Letting .U = log y / log Z, we see that U − U =

.

δU log U  δ(1 − )U. log Z

Hence, .U  (1 + δ(1 − ))U and .U log U  (1 + δ(1 − ))U log U . Thus, .

2 (y , Z) y 2 (y , Z) ≥ ≥ exp(U log U − U log U + o(U log U )), 2 (y, Z) (y, Z) y

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which is at least .exp((δ + o(1))U log U ), and hence (eventually) larger than 2. Therefore,  .

m∈(y,x 1/2 ] m Z-smooth

μ2 (m) ≥ m

 m∈(y,y ] m Z-smooth

≥

μ2 (m) 1 ≥ (2 (y , Z) − 2 (y, Z)) m y

1 2 (y , Z) ≥ exp(−(1 + o(1))U log U ). 2y

 Since .U  (1 + δ(1 − ))U and .U ∼ u, we conclude that . m∈(y,x 1/2 ], m Z-smooth μ2 (m)/m is eventually larger than .exp(−(1 + δ)u log u). But .δ > 0 was arbitrary, and so  .

m∈(y,x 1/2 ] m Z-smooth

μ2 (m) ≥ exp(−(1 + o(1))u log u), m

as desired. A more careful argument is required when .y ≤ (log log x)log log log log x . We consider .n ≤ x of the form .n = mq, where m is a divisor of L from the interval .(y, x 1/2 ], q has all prime factors exceeding Z, and .L | σ (q). Each such n has .gcd(n, σ (n)) ≥ m > y, and our earlier arguments show that the number of such .n ≤ x is .

∼ e−γ

x log log log x

 m|L y Z (forcing .e > 1). Thus .m = m0 m1 , where .m0 is squarefull and larger than Z, and .m1 is Z-smooth. But the reciprocal sum of all such m is at most  .

m0 >Z squarefull

1 m0

 m1 Z-smooth

⎛ 1 =⎝ m1



≤Z

⎞ (1 − 1/ )−1 ⎠

 m0 >Z squarefull

1 log Z .  √  (log log x)1/3 Z

1 m0

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Turning to the analogous sum over Z-smooth m, we have that  .

m Z-smooth yZ

 p≡−1 (mod ) p≤x

 log log x 1 x x ,  2 p

exp((log log log x)2/3 )

>Z

which is dominated by .x exp(−u log u) in this range of u. But if .gcd(n, σ (n)) is Z -smooth, then n has a .Z -smooth divisor exceeding y. In this situation we can invoke the following estimate of Tenenbaum, which is a special case of Exercise 293 on pp. 554–555 of [19]. (Alternatively, we could apply [18, Lemme 3].)

.

Proposition 3.1 Let .X ≥ Y ≥ Y ≥ 2. The number of .n ≤ X whose Y -smooth part exceeds .Y is .

 x exp(−V log V ) + x/Y 1/2 ,

where .V := log Y / log Y . Taking .X = x, .Y = Z , and .Y = y in Proposition 3.1 bounds the number of n as above as . x exp(−(1 + o(1))u log u) + x/y 1/2 . Since .y = (log log x)u > u2u , this is at most .x exp(−(1 + o(1))u log u). So we have the upper bound (14) in this case, with the better constant 1 replacing . 13 . Suppose instead that .u > (log log log x)1/2 . For this case we need the following lemmas, proved in [14]. Lemma 3.1 (See [14, Lemma 2.4]) There is an absolute constant C such that the following holds. For each .x ≥ 1 and each squarefree number d, the number of squarefree .n ≤ x for which .d | σ (n) is at most .

x (Cω(d) log log max{3, x})ω(d) . ϕ(d)

Lemma 3.2 (See [14, Theorem 1.3]) For all .T ≥ 3, we have  .

√

gcd(n, σ (n)) ≤ T 1+O(1/

log log T )

.

n≤T

Remark We do not require the full force of Lemma 3.2; we could get by with an upper bound of .T 1+o(1) for the same sum restricted to squarefree n. Such an estimate

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can be shown in a simpler way (cf. the proof of the upper bound in Theorem 11 of [7]). .

Let .d = gcd(n, σ (n)), and observe that if .n = de, then .d | σ (d)σ (e), and so d gcd(d,σ (d)) | σ (e). We take cases according to the size of .w := ω(d/ gcd(d, σ (d))). Suppose first d that .w ≤ 13 u. Since .n = de with .e ≤ x/d and . gcd(d,σ (d)) | σ (e), the number of possibilities for n given d is bounded by

1u 3 1 x C · u log log x . . d · ϕ(d/ gcd(d, σ (d))) 3

 1u 2 2 3 Since .u ≤ (log log x)1− , we have . C · 13 u log log x < (log log x) 3 u = y 3 . d/ gcd(d,σ (d)) o(1) , we find that Now we sum on d. Using that . ϕ(d/ gcd(d,σ (d)))  log log x = y

 .

d>y squarefree

 gcd(d, σ (d)) 1 ≤ y o(1) = y −1+o(1) , d · ϕ(d/ gcd(d, σ (d))) d2 d>y

where the final sum on d was estimated using Lemma 3.2 and partial summation. So this case contributes at most .xy −1/3+o(1) values of n. Since .y = (log log x)u > uu , this is acceptable. Now suppose that .w > 13 u. We can assume that d (and hence also .d/ gcd(d, σ (d))) is y-smooth. Indeed, a now familiar argument shows that the number of squarefree .n ≤ x with .gcd(n, σ (n)) divisible by a prime exceeding y is is . x log log x/y, which is .x/y 1+o(1) (remembering that .log log x = y 1/u ) and thus acceptable. Viewing .d = d/ gcd(d, σ (d)), the number of remaining n can be bounded, in terms of an arbitrary parameter .t > 1, by x



.

d squarefree y-smooth ω(d )> 13 u

1 ≤ xt −u/3 d

 d squarefree y-smooth



t t ω(d ) −u/3 1 + = xt d p p≤y

 u  ≤ x exp − log t + t log log y + O(t) . 3

(15)

We choose .t = u/ log u. Observe that .log log y = log u + log log log log x < 3 log u (in this range of y), so that .

(log u)2 1 u log t   log u, t log log y log log y 3

On the Greatest Common Divisor of a Number and Its Sum of Divisors, II

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which tends to infinity. Thus, .t log log y = o(u log t), rendering the upper bound in (15) of size at most .x exp(−( 13 + o(1))u log u). This completes the proof of (14). To transition from .E ∗ to E, let n be any integer in .[1, x] with .gcd(n, σ (n)) > y, and write .n = n0 n1 , where .n0 is squarefree, .n1 is squarefull, and .gcd(n0 , n1 ) = 1. Then (for x large) y < gcd(n, σ (n)) ≤ gcd(n0 , σ (n0 )) gcd(n0 , σ (n1 )) gcd(n1 , σ (n0 )σ (n1 ))

.

≤ gcd(n0 , σ (n0 ))σ (n1 )n1 ≤ 2 gcd(n0 , σ (n0 ))n21 log log x. (16) Thus, either (a) .n1 > y 2/7 / log log x or (b) .n1 ≤ y 2/7 / log log x and 3/7 . The number of .n ≤ x in case (a) is .O(x(log log x)1/2 /y 1/7 ), .gcd(n0 , σ (n0 )) > y which is at most .x/y 1/7+o(1) and acceptable for (3). The number of n in case (b) is bounded by 

E ∗ (x/n1 , y 3/7 ).

.

n1 ≤y 2/7 , n1 squarefull

As .log (y 3/7 )/ log log(x/n1 ) ∼ 37 u, our bounds on .E ∗ yield E ∗ (x/n1 , y 3/7 ) ≤

.

x exp(−(1/7 + o(1))u log u), n1

 uniformly for squarefull .n1 ≤ y 2/7 . Since . 1/n1  1, the upper bound in (3) follows. Remark A more elaborate version of the argument going from .E ∗ to E would lead to an improvement (increase) of the constant .1/7 in (4). We have not pursued this, since we suspect that (14) itself is not optimal.

3.4 Upper Bound in Theorem 1.1 We first argue that the bound (14) can be improved to E ∗ (x, y) ≤ x exp (− (1 + o(1)) u log u)

.

(17)

when .y ≤ exp((log log x)o(1) ). (Of course, we continue to assume that .u = log y/ log log log x → ∞.) We will suppose that .u > (log log log x)1/2 , as our earlier arguments already establish (17) in the complementary range. We treat this range of u by the same the method used in proving Theorem 1.2. However, instead of splitting the possible values of .w = ω(d/ gcd(d, σ (d))) at

282

P. Pollack

we split at .(1 − η)u, where .η > 0 is small and fixed. Keeping in mind that o(1) , we see that .(C(1 − η)u log log x)(1−η)u ≤ y 1−η+o(1) , and then .u ≤ (log log x) that the number of n corresponding to some .w ≤ (1 − η)u is at most .xy −η+o(1) . Since .u = (log log x)o(1) , eventually .y −η+o(1) < y −η/2 = (log log x)−uη/2 < exp(−u log u), so that there are fewer than .x exp(−u log u) values of n of this kind. On the other hand, there are at most .x exp(−(1 − η + o(1))u log u) numbers n corresponding to some .w ≥ (1 − η)u. This proves (17) with .1 + o(1) replaced by .1 − η + o(1). Since .η can be taken arbitrarily small, (17) follows. Now suppose that .n ≤ x is not necessarily squarefree and that .gcd(n, σ (n)) > y. Again, we fix a small real number .η > 0. Write .n = n0 n1 where .n0 is squarefree, .n1 is squarefull, and .gcd(n0 , n1 ) = 1. We can assume .n1 ≤ y η / log log x, since the number of .n ≤ x with squarefull component exceeding .y η / log log x is .O(x(log log x)1/2 /y η/2 ) and (eventually) η/2 /(log log x)1/2 > y η/3 = (log log x)uη/3 > uu . So from (16), .y

.

1 3 u,

d0 := gcd(n0 , σ (n0 )) > y 1−2η .

.

Thus, given .n1 , the number of corresponding .n0 is at most E ∗ (x/n1 , y 1−2η ) ≤

.

x exp(−(1 − 2η + o(1))u log u), n1

uniformly in .n1 ≤ y η . Summing on squarefull .n1 , and keeping mind that .η can be taken arbitrarily small, we obtain the upper bound on .E(x, y) claimed in Theorem 1.1. Acknowledgments The author is supported by NSF award DMS-2001581. He thanks Kevin Ford and Carl Pomerance for many enlightening discussions on related topics, and he thanks the referee and Paco Adajar for helpful comments. This paper was finished while the author enjoyed the hospitality of the Budapest Semesters in Mathematics Program as a ‘Mathematician in Residence’, jointly with Enrique Treviño.

References 1. E.R. Canfield, P. Erd˝os, C. Pomerance, On a problem of Oppenheim concerning “factorisatio numerorum”. J. Number Theory 17, 1–28 (1983) 2. H. Davenport, Über numeri abundantes. S.-Ber. Preuß. Akad. Wiss., math.-nat. Kl., 830–837 (1933) 3. N.G. de Bruijn, On the number of positive integers ≤ x and free prime factors > y. II. Nederl. Akad. Wetensch. Proc. Ser. A 69=Indag. Math. 28, 239–247 (1966) 4. R. de la Bretèche, G. Tenenbaum, Sur les lois locales de la répartition du k-ième diviseur d’un entier. Proc. Lond. Math. Soc. (3) 84, 289–323 (2002) 5. P. Erd˝os, Some asymptotic formulas in number theory. J. Indian Math. Soc. (N.S.) 12, 75–78 (1948)

On the Greatest Common Divisor of a Number and Its Sum of Divisors, II

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6. P. Erd˝os, On perfect and multiply perfect numbers. Ann. Mat. Pura Appl. (4) 42, 253–258 (1956) 7. P. Erd˝os, F. Luca, C. Pomerance, On the proportion of numbers coprime to a given integer, in Anatomy of Integers, CRM Proceedings. Lecture Notes, vol. 46 (American Mathematical Society, Providence, RI, 2008), pp. 47–64 8. B. Hornfeck, Zur Dichte der Menge der vollkommenen Zahlen. Arch. Math. (Basel) 6, 442– 443 (1955) 9. B. Hornfeck, E. Wirsing, Über die Häufigkeit vollkommener Zahlen. Math. Ann. 133, 431–438 (1957) 10. H.-J. Kanold, Über die Dichten der Mengen der vollkommenen und der befreundeten Zahlen. Math. Z. 61, 180–185 (1954) 11. H.-J. Kanold, Eine Bemerkung über die Menge der vollkommenen Zahlen. Math. Ann. 131, 390–392 (1956) 12. H.-J. Kanold, Über die Verteilung der vollkommenen Zahlen und allgemeinerer Zahlenmengen. Math. Ann. 132, 442–450 (1957) 13. M. Naimi, Les entiers sans facteurs carré ≤ x dont leurs facteurs premiers ≤ y, in Groupe de travail en théorie analytique et élémentaire des nombres, 1986–1987. Publ. Math. Orsay, vol. 88 (Univ. Paris XI, Orsay, 1988), pp. 69–76 14. P. Pollack, On the greatest common divisor of a number and its sum of divisors. Michigan Math. J. 60, 199–214 (2011) 15. P. Pollack, A. Singha Roy, Powerfree sums of proper divisors. Colloq. Math. 168, 287–295 (2022) 16. C. Pomerance, On the distribution of amicable numbers. J. Reine Angew. Math. 293(294), 217–222 (1977) 17. E.J. Scourfield, An asymptotic formula for the property (n, f (n)) = 1 for a class of multiplicative functions. Acta Arith. 29, 401–423 (1976) 18. G. Tenenbaum, Sur la probabilité qu’un entier possède un diviseur dans un intervalle donné. Compos. Math. 51, 243–263 (1984) 19. G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3rd edn. Graduate Studies in Mathematics, vol. 163 (American Mathematical Society, Providence, RI, 2015) 20. B. Volkmann, Ein Satz über die Menge der vollkommenen Zahlen. J. Reine Angew. Math. 195 (1955), 152–155 (1956) 21. E. Wirsing, Bemerkung zu der Arbeit über vollkommene Zahlen. Math. Ann. 137, 316–318 (1959)

Permutations with Arithmetic Constraints Carl Pomerance

In memory of Eduard Wirsing (1931–2022)

Abstract Let .Slcm (n) denote the set of permutations .π of .[n] = {1, 2, . . . , n} such that .lcm[j, π(j )] ≤ n for each .j ∈ [n]. Further, let .Sdiv (n) denote the number of permutations .π of .[n] such that .j | π(j ) or .π(j ) | j for each .j ∈ [n]. Clearly .Sdiv (n) ⊂ Slcm (n). We get upper and lower bounds for the counts of these sets, showing they grow geometrically. We also prove a conjecture from a recent paper on the number of “anti-coprime” permutations of .[n], meaning that each .gcd(j, π(j )) > 1 except when .j = 1. Keywords Permutations · Arithmetic constraints

1 Introduction Recently in [7] some permutation enumeration problems with an arithmetic flavor were considered. In particular, one might count permutations .π of .[n] = {1, 2, . . . , n} where each .gcd(j, π(j )) = 1 and also permutations .π where each .gcd(j, π(j )) > 1 except for .j = 1. It was shown in [7] that the coprime count is between .n!/c1n and .n!/c2n for all large n, where .c1 = 3.73 and .c2 = 2.5. Shortly after, Sah and Sawhney [9] showed that there is an explicit constant .c0 = 2.65044 . . . with the count of the shape .n!/(c0 + o(1))n as .n → ∞. The “anti-coprime” count was shown in [7] to exceed .n!/(log n)(α+o(1))n as .n → ∞, where .α = e−γ , with .γ Euler’s constant. It was conjectured in [7] that this lower bound is sharp, which we will prove here.

C. Pomerance () Mathematics Department, Dartmouth College, Hanover, NH, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Maier et al. (eds.), Number Theory in Memory of Eduard Wirsing, https://doi.org/10.1007/978-3-031-31617-3_19

285

286

C. Pomerance

There are several papers in the literature that have considered the divisibility graph on .[n] where .i = j are connected by an edge if i divides j or vice versa, and the closely related lcm graph, where edges correspond to .lcm[i, j ] ≤ n. In particular, it was shown in [6] that the length of the longest simple path in such graphs is .o(n), and this has been improved to order-of-magnitude .n/ log n, see Saias [10] for a recent paper on the topic. One might also consider permutations of .[n] compatible with these graphs. Let .Sdiv (n) denote the set of permutations .π of n such that for each .j ∈ [n], either .j | π(j ) or .π(j ) | j . Further, let .Slcm (n) denote the set of permutations .π of .[n] such that for each .j ∈ [n], .lcm[j, π(j )] ≤ n. Clearly, .Sdiv (n) ⊂ Slcm (n). There is a small literature on these topics. In particular, counts for .#Sdiv (n) are on OEIS [5] (due to Heinz and Farrokhi), which we reproduce here, together with new counts for .#Slcm (n). Table 1 suggests that .#Slcm (n) > #Sdiv (n) > 2n for n large, and that there may be a similar upper bound. In this note we will prove that .(#Sdiv (n))1/n is bounded above 1 and .(#Slcm (n))1/n is bounded below infinity. We conjecture they tend to limits, but we lack the numerical evidence or heuristics to suggest values for these limits.1 We will also show that .#Slcm (n)/#Sdiv (n) tends to infinity geometrically. One might also ask for the length of the longest cycle among permutations in .Sdiv (n) or in .Slcm (n). This seems to be only slightly less (if at all) than the length of the longest simple chain in the divisor graph or lcm graph on .[n] mentioned above. Other papers have looked at tilings of .[n] with divisor chains, for example see [4]. This could correspond to asking about the cycle decomposition for permutations in .Sdiv (n) or in .Slcm (n). We mention the paper [2] of Erd˝os, Freud, and Hegyvári where some other arithmetic problems connected with integer permutations are discussed. Finally, we note the recent paper [1] which also has a similar flavor.

2 An Upper Bound for #Slcm (n) Theorem 2.1 We have #Slcm (n) ≤ e2.61n for all large n. Proof Let n be large. For j ∈ [n], let N(j ) denote the number of j  ∈ [n] with lcm[j, j  ] ≤ n. This condition can be broken down as follows: lcm[j, j  ] ≤ n if and only if there are integers a, b, c with j = ab,

.

j  = bc,

gcd(a, c) = 1,

abc ≤ n.

(1)

That is, N (j ) is the number of triples a, b, c with j = ab satisfying (1). Since a | j , b = j/a, and c ≤ n/j , we have N(j ) ≤ τ (j )n/j , where τ is the divisor function (which counts the number of positive divisors of its argument). For π ∈ Slcm (n), the

1 This

conjecture was very recently proved by McNew, see arXiv:2207.09652 [math.NT].

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Table 1 Counts for .Sdiv (n) and .Slcm (n) and their nth roots n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

.#Sdiv (n)

.(#Sdiv (n))

1 2 3 8 10 36 41 132 250 700 750 4010 4237 10,680 24,679 87,328 90,478 435,812 449,586 1,939,684 3,853,278 8,650,900 8,840,110 60,035,322 80,605,209 177,211,024 368,759,752 1,380,348,224 1,401,414,640 8,892,787,136 9,014,369,784 33,923,638,848 59,455,553,072 126,536,289,568 207,587,882,368

1.0000 1.4142 1.4422 1.6818 1.5849 2.8272 1.6998 1.8411 1.8469 1.9254 1.8254 1.9965 1.9011 1.9398 1.9626 2.0362 1.9569 2.0573 1.9839 2.0625 2.0588 2.0669 2.0046 2.1091 2.0714 2.0761 2.0757 2.1205 2.0673 2.1460 2.0947 2.1334 2.1208 2.1210 2.1055

1/n

.#Slcm (n)

.(#Slcm (n))

1 2 3 8 10 56 64 192 332 1184 1264 12,192 12,872 37,568 100,836 311,760 322,320 2,338,368 2,408,848 14,433,408 32,058,912 76,931,008 78,528,704 919,469,408 1,158,792,224 2,689,828,672 4,675,217,824 21,679,173,184 21,984,820,864 381,078,324,992 386,159,441,600 1,202,247,415,040

1.0000 1.4142 1.4422 1.6818 1.5849 1.9560 1.8114 1.9294 1.9060 2.0292 1.9142 2.1903 2.0708 2.1221 2.1556 2.2048 2.1087 2.2585 2.1671 2.2802 2.2773 2.2828 2.2043 2.3631 2.3044 2.3051 2.2811 2.3396 2.2731 2.4324 2.3646 2.3851

1/n

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number of possible values for π(j ) is at most N(j ), so we have #Slcm (n) ≤



.



N(j ) ≤

j ∈[n]

τ (j )n/j.

(2)

j ∈[n]

This quickly leads to an estimate for #Slcm (n) that is of the form n!o(1) as n → ∞, but to do better we will need to work harder. In particular we use a seemingly trivial property of permutations: they are one-to-one. In particular, there are not many values of j with π(j ) small, since there are not many small numbers. This thought leads to versions of (2) where τ is replaced with a restricted divisor function that counts only small divisors. Let k = 30. We partition the interval (0, n] into subintervals as follows. Let i J0 = (n/k, n]. Let i0 be the largest i such that L := k 2 ≤ log n, so that (log n)1/2 < i i−1 L ≤ log n. For i = 1, . . . , i0 , let Ji = (n/k 2 , n/k 2 ], and let Ji0 +1 = (0, n/L]. For π ∈ Slcm (n) we have sets Xi , Yi as follows: i

Xi :={j ∈ Ji : π(j ) > n/k 2 },

.

Yi :={j > n/k

2i−1

0 ≤ i ≤ i0 ,

: π(j ) ∈ Ji }, 1 ≤ i ≤ i0 + 1.

These sets depend on the choice of π , but the number of choices for the sets Yi is not so large. We begin by counting the number of possibilities for the sequence of sets Y1 , . . . , Yi0 +1 . Since π is a permutation it follows that yi := #Yi is at most the number of i−1 i−1 integers in Ji , so that yi ≤ n/k 2 . The number of subsets of (n/k 2 , n] of cardinality ≤ yi is less than  n .

u≤yi

   n  n 2nyi ≤2 ≤ ≤ exp (2i−1 log k + 1) , i−1 u yi yi ! k2

for n sufficiently large, using the inequality 2/j ! < (e/j )j for j ≥ 3. Multiplying these estimates we obtain that the number of choices for a sequence of sets {Yi } as described is .

≤ exp(0.1554n)

(3)

for all sufficiently large n. Fix now a specific sequence of sets {Yi }, which then determines a complementary sequence of sets {Xi } with Xi = Ji \Yi+1 . We will give the set X0 special treatment, so for now, assume that 1 ≤ i ≤ i0 . For j ∈ Xi , the number of possible choices j  to which j may be mapped by a permutation in Slcm (n) (with sequence of sets {Yi }) is at most the number of choices for a, c as in (1). Here c ≤ n/j and a | j with i a ≤ n/j  < k 2 . Let τz (m) be the number of divisors of m that are < z. With this

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289

notation, the number of choices for π ∈ Slcm (n) restricted to Xi is at most  j ∈Xi

τk 2i (j )

n ≤ j



τk 2i (j ) i−1

j ≤n/k 2

n j

⎛

.

⎜ ≤⎝

⎞n/k 2i−1



1

n/k 2

i−1



⎟ τk 2i (j )⎠

i−1 j ≤n/k 2

nn/k

(4)

2i−1

n/k 2

i−1

!

,

by the AM-GM inequality (the  arithmetic mean geometric mean inequality). Since the harmonic sum d 2(1−o(1))(log x)/ log log x . n≤x

Theorem 1.2 .

max |g(n)| > 2(2/3−o(1))(log x)/ log log x . n≤x

Conjecture .

max |g(n)| > 2(1−o(1))(log x)/ log log x . n≤x

We give some heuristics to support this conjecture in the next section. It is worthwhile to recall that the typical values of these functions are very small. With the notation δ(t) = d{n : M(n, t) = 0}

.

Erd˝os and Hall [2] proved that   δ(t) = O (log t)−c , c = 0.0579 . . . .

.

Large Subsums of the Möbius Function

301

Their proof can easily be adapted to show that .

d {n : g(n) = 0} = 1.

Probably the estimate |{n ≤ x : g(n) = 0}| < x/(log x)c

.

also holds with some positive c, but I have not checked the details. Maier [4] proved that typically .f (n) is between two powers of .log log n. The best exponents, due to Maier-Tenenbaum [5] are that for almost all n (log log n)c1 −ε < f (n) < (log log n)c2 +ε ,

.

c1 = 0.28754..., .c2 = log 2. For the square mean, de la Bretèche, Dress, Tenenbaum [6] proved that

.

.

1 |M(n, t)|2 = L + o(1) x n≤x

with some constant L uniformly as long as .t → ∞ and .t = o(x). Conjecture  .

g(n)2 = Lx + o(x).

n≤x

I can prove the weaker estimate  .

g(n)2 = O(x).

n≤x

2 Proof of Theorem 1.1 Proof We use the Dirichlet series of the Möbius function for divisors of n at imaginary values: D(α) =



.

d|n



n

μ(d)d iα = −iα 1

M(n, t)t iα−1 dt,

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I. Z. Ruzsa

whence  .

n

|D(α)| < |α| 1

|M(n, t)| dt < |α|f (n) log n. t

So to find a large value of f (n) it is sufficient to find a single large value of D for a not too large α. Write D(α) as an Euler product: D(α) =

.

     α log p 1 − piα = niα/2 p−iα/2 − piα/2 = niα/2 (−2)k sin 2 p|n

p|n

p|n

with k = ω(n), so  α log p . |D(α)| = 2 sin 2 . k

p|n

For n we shall take the product of primes of an interval (u, 2u) and put α = π/(log u). For each prime we have .

α log p π π < < 2 2 2

 log 2 π 1+ = +ε log u 2

with ε = (log 2)/ log u, hence each sinus is > 1 − ε and |D(α)| > 2(1−ε)k . The product of primes in the interval (u, 2u) is 

p ≤ e(2u)−(u) = eu(1+o(1)) ,

.

u (u + v)m , that is,  .

1+

v m < u. u

m  As . 1 + uv ≈ emv/u , this is about the same as .mv < u log u. We need .k = 2m + 1 primes. The average distance between primes around u is 2 2 .log u. This means .v ∼ 2m log u, or .u > 2m ∼ k /2. To ensure .n < x the condition 2 k is .(k /2) < x, which yields k∼

.

1 log x . 2 log log x

This simple construction gives Theorem 2 with 1/2 in the place of 2/3. The improvement, to be detailed in the next section, is based on the observation that we √ do not actually need that all divisors composed of at most .m primes should be .< n, only that most of them be.

304

I. Z. Ruzsa

4 Proof of Theorem 1.2 Write .n = p1 . . . pk , .k = 2m + 1, .u = p1 < . . . < pk = u + v. Our choice will be

k∼

.

 log x 2 −ε 3 log log x

and .u ∼ k 3/2 log k. Around u the typical distance between primes is .log u, we expect .v ∼ k log u. To show that this is indeed the case we recall Heath-Brown’s estimate for primes in intervals [3] by which .π(v + w) − π(v) ∼ w/ log w as long 2/3 as .u7/12+ε < w < u and in our case the interval is longer than √ .u . We estimate the ratio of products of m terms below . n. Put .a = (log n)/k, .log pi = a + bi . Clearly log u < a < log(u + v).

.

For .d|n, .ω(d) = j we have .log d = j a+ sum of some .bi . √ If we can ensure that from the .2k sum formed by the numbers .bi only .o(2k / k) are .> a/2, then



k  √ k−1 2 ∼ ±c2k / k +o √ .g(n) = (−1) m k m

will hold as wanted. We expect the sum of the .bi to have approximately a normal distribution. To make this exact we use Bernstein’s inequality in the following form. Let .ξ1 , . . . , ξk be independent random variables with zero mean and .|ξi | ≤ M. Then .

Prob





 −t 2 /2 ξi ≥ t ≤ exp , E(ξi2 ) + Mt/3 

see e.g. https://en.wikipedia.org/wiki/Bernstein_inequalities_(probability_theory). We apply the values .±b probability  1/2 each. (Note

this for .ξi assuming

i /2 with that . bi = 0, hence . i∈I bi = (1/2) b − b i , so subsums of .bi i∈I i i ∈I / have the same distribution as . ξi .) Clearly |bi | < log

.

u+v < v/u, u

so we can set .M = v/(2u). We put .t = a/2, and estimate .E(ξi2 ) by .M 2 .

Large Subsums of the Möbius Function

305

We claim that with this choice of the arguments (for sufficiently large value of the parameters) we have

.

t 2 /2 > log k, E(ξi2 ) + Mt/3

hence the probability of large values will be .< 1/k, that is, the number of subsums with an unusual large contribution from the .bi is .O(2k /k). After substituting .t = a/2, and estimating .E(ξi2 ) by .M 2 we obtain that it is sufficient to show   2 2 .8 log k kM + aM/6 < a . (3) To see this we express these quantities asymptotically by k. We have u ∼ k 3/2 log k, v ∼ k log u ∼

.

M=

.

3 k log k, 2

3 v ∼ √ , 2u 4 k

hence 8(log k)kM 2 ∼

.

9 log k = o(a 2 ), 2

as .a > log u > log k, so the first summand in (3) is small enough. For the second observe that

M log k ∼

.

3 log k √ = o(1), 4 k

so .aM log k = o(a 2 ) as well. This concludes the proof of (3).

References 1. P. Erd˝os, On a problem in elementary number theory. Math. Student 17, 32–33 (1949) 2. P. Erd˝os, R.R. Hall, On the Möbius function. J. Reine Angew. Math. 315, 121–128 (1980) 3. D.R. Heath-Brown, The number of primes in a short interval. J. Reine Angew. Math. 389, 22–63 (1988) 4. H. Maier, On the Möbius function. Trans. Am. Math. Soc. 301, 649–664 (1987) 5. H. Maier, G. Tenebaum, On the normal concentration of divisors 2. Math. Proc. Camb. Philos. Soc. 147, 593–614 (2009) 6. G. Tenenbaum R. de la Bretèche, F. Dress, Remarques sur une somme liée à la fonction de Möbius. Mathematika 66, 416–421 (2020)

The a-Points of the Riemann Zeta-Function and the Functional Equation Athanasios Sourmelidis, Jörn Steuding, and Ade Irma Suriajaya

Dedicated to the memory of Professor Eduard Wirsing

Abstract We prove an equivalent of the Riemann hypothesis in terms of the functional equation (in its asymmetrical form) and the a-points of the zeta-function, i.e., the roots of the equation .ζ (s) = a, where a is an arbitrary fixed complex number. Keywords Riemann zeta-function · Riemann hypothesis · a-points · Functional equation

1 Motivation and Statement of the Main Results Let .s = σ + it be a complex variable. The Riemann zeta-function .ζ is for .σ > 1 defined by ζ (s) =

.

 (1 − p−s )−1 , p

where the product is taken over all prime numbers p, and by analytic continuation elsewhere except for a simple pole at .s = 1. The Euler product representation above

A. Sourmelidis Institute of Analysis and Number Theory, TU Graz, Graz, Austria e-mail: [email protected] J. Steuding Department of Mathematics, Würzburg University, Würzburg, Germany e-mail: [email protected] A. I. Suriajaya () Faculty of Mathematics, Kyushu University, Fukuoka, Japan e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Maier et al. (eds.), Number Theory in Memory of Eduard Wirsing, https://doi.org/10.1007/978-3-031-31617-3_21

307

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A. Sourmelidis et al.

indicates the relevance of .ζ for the distribution of prime numbers. The yet unproven Riemann hypothesis claims that all nontrivial (non-real) zeros .ρ = β + iγ of the Riemann zeta-function .ζ lie on the critical line .1/2 + iR. This distribution of zeros would lead to the least possible error term in the prime number theorem. The Chebyshev function is defined by ψ(x) =



.

(n),

n≤x

where . denotes the von Mangoldt-function that counts prime powers .n = pk with logarithmic weight .log p. The prime number theorem is the asymptotic formula .ψ(x) ∼ x, and the Riemann hypothesis is equivalent to ψ(x) = x + O (x 1/2+ ),

.

(1)

where here and in the sequel . > 0 is arbitrary; this equivalence was first proved by Helge von Koch [19] and relies on the explicit formula from Bernhard Riemann’s path-breaking memoir [15]. Another more analytic aspect is the distribution of values of .ζ . Given a complex number a, the roots of the equation .ζ (s) = a are called a-points; we denote these roots in the right half-plane by .ρa = βa + iγa and their count is very similar to the number of nontrivial zeros (i.e., the case .a = 0). It was Edmund Landau who suggested in his invited lecture [12] at the International Congress of Mathematicians in Cambridge 1912 to study the distribution of a-points with the words: “The points at which an analytic function is equal to 0 are very important; but equally interesting are the points at which it assumes a certain value a.”.1 The so-called Nevanlinna theory, or value distribution theory, developed by Rolf Nevanlinna a little later, takes up this idea. In this note we investigate the a-points of .ζ in the context of the functional equation, that is ζ (s) = (s)ζ (1 − s),

(2)

(s) := 2(2π )−s sin π2s (1 − s).

(3)

.

where .

Note that we could as well consider other functions satisfying a similar functional equation, e.g. Dirichlet L-functions. However, for the sake of simplicity, we restrict to the case of .ζ .

1 The authors’ translation of the German original text: “Es sind bei einer analytischen Funktion die Punkte, an denen sie 0 ist, zwar sehr wichtig; ebenso interessant sind aber die Punkte, an denen sie einen bestimmten Wert a annimmt.”

The a-Points of the Riemann Zeta-Function

309

Our first result deals with an asymptotic formula for . at the a-points. Theorem 1.1 Let a be an arbitrary fixed complex number. Then, as .T → ∞,  .

βa ≥0, 0