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Carl Pomerance · Michael Th. Rassias Editors in Honor of Helmut Maier's Carl Pomerance · Michael Th. Rassias Editors Analytic Number Theory In Honor of Helmut Maier’s 60th Birthday Analytic Number Theory Professor Helmut Maier Carl Pomerance • Michael Th. Rassias Editors Analytic Number Theory In Honor of Helmut Maier’s 60th Birthday 123 Editors Carl Pomerance Michael Th. Rassias Department of Mathematics Department of Mathematics Dartmouth College ETH-Zürich Hanover, NH, USA Zürich, Switzerland Department of Mathematics Princeton University Princeton, NJ, USA ISBN 978-3-319-22239-4 ISBN 978-3-319-22240-0 (eBook) DOI 10.1007/978-3-319-22240-0 Library of Congress Control Number: 2015947785 Mathematics Subject Classification (2010): 11-Axx, 11-Bxx, 11-Lxx, 11-Mxx, 11-Nxx, 11-Pxx, 11-Yxx, 26-Dxx, 40-XX, 41-XX Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved 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, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com) Preface This book presents a series of research and expository articles under the broad banner of Analytic Number Theory. Written by eminent mathematicians from the international mathematical community, the chapters in this book are dedicated to the 60th birthday of Professor Helmut Maier. His own research has been groundbreaking and deeply influential, especially in the problem of gaps between consecutive primes. Helmut Maier was born in Geislingen/Steige, Germany, on October 17, 1953. He graduated with a Diploma in Mathematics from the University of Ulm in 1976, under the supervision of Professor H.E. Richert. After a period as a scientific employee at the University of Ulm, he started his doctoral studies at the University of Minnesota in 1979. He received his Ph.D. in 1981 with the thesis entitled Some results on prime numbers based on the application of sieve theory, under the supervision of Professor J. Ian Richards. The thesis was an extension of his paper Chains of large gaps between consecutive primes, which appeared in Advances in Mathematics, 39 (1981), 257–269. In this paper, Maier applied for the first time what is now called Maier’s Matrix Method. This method later on led him and other mathematicians to the discovery of unexpected irregularities in the distribution of prime numbers. The Matrix Method was also applied to other questions like irreducible polynomials and consecutive primes in the same residue class modulo an integer. After postdoctoral positions at the University of Michigan and the Institute for Advanced Study, Princeton, Maier then took a permanent position at the University of Georgia. During this period, he showed that the usual formulation of the Cramér model for the distribution of primes is wrong, a completely unexpected result. With C. Pomerance, he did research on the values of Euler’s '-function and large gaps between primes. One theme in this period was Professor Maier’s work on the size of the coefficients of cyclotomic polynomials, done partially in collaboration with S. Konyagin and E. Wirsing. He also collaborated with H.L. Montgomery on the size of the sum of the Möbius function under the assumption of the Riemann Hypothesis. Maier and G. Tenenbaum in joint work investigated the sequence of divisors of integers, solving the famous propinquity problem of Erdos.˝ v vi Preface Since 1993, Maier is a Professor at the University of Ulm, Germany. Lately, M.Th. Rassias and Maier worked on problems related to the Estermann zeta function and the Nyman–Beurling criterion on the Riemann Hypothesis involving the distribution of certain cotangent sums. In their latest work, they studied prime avoidance of perfect powers of prime numbers. Other collaborators of Professor Maier include P. Erdos,˝ J. Friedlander, A. Granville, D. Haase, A.J. Hildebrand, M.L. Lapidus, J.W. Neuberger, A. Sankaranarayanan, A. Sárközy, and C.L. Stewart. We are grateful to all of the mathematicians who participated in this publication. We also wish to express our thanks to Springer for undertaking the publication of this book. Hanover, NH, USA Carl Pomerance Princeton, NJ, USA Michael Th. Rassias Contents CM-Points on Straight Lines ................................................... 1 Bill Allombert, Yuri Bilu, and Amalia Pizarro-Madariaga Maass Waveforms and Low-Lying Zeros ...................................... 19 Levent Alpoge, Nadine Amersi, Geoffrey Iyer, Oleg Lazarev, Steven J. Miller, and Liyang Zhang Théorème de Jordan Friable.................................................... 57 Régis de la Bretèche et Gérald Tenenbaum On Conjectures of T. Ordowski and Z.W. Sun Concerning Primes and Quadratic Forms ................................................... 65 Christian Elsholtz and Glyn Harman Large Gaps Between Consecutive Prime Numbers Containing Perfect Powers .................................................................... 83 Kevin Ford, D.R. Heath-Brown, and Sergei Konyagin On the Parity of the Number of Small Divisors of n ......................... 93 Kevin Ford, Florian Luca, Carl Pomerance, and Jeffrey Shallit Counting Primes in Arithmetic Progressions ................................. 101 John B. Friedlander Limit Points of the Sequence of Normalized Differences Between Consecutive Prime Numbers ......................................... 115 Daniel A. Goldston and Andrew H. Ledoan Spirals of the Zeta Function I................................................... 127 Steven M. Gonek and Hugh L. Montgomery Best Possible Densities of Dickson m-Tuples, as a Consequence of Zhang–Maynard–Tao......................................................... 133 Andrew Granville, Daniel M. Kane, Dimitris Koukoulopoulos, and Robert J. Lemke Oliver vii viii Contents A Note on Helson’s Conjecture on Moments of Random Multiplicative Functions......................................................... 145 Adam J. Harper, Ashkan Nikeghbali, and Maksym Radziwiłł Large Values of the Zeta-Function on the Critical Line ..................... 171 Aleksandar Ivic´ A Note on Bessel Twists of L-Functions........................................ 195 J. Kaczorowski and A. Perelli The Sound of Fractal Strings and the Riemann Hypothesis ................ 201 Michel L. Lapidus Sums of Two Squares in Short Intervals....................................... 253 James Maynard Infinite Sumsets with Many Representations ................................. 275 Melvyn B. Nathanson On the Ratio of Consecutive Gaps Between Primes .......................... 285 János Pintz Remarks on Fibers of the Sum-of-Divisors Function ........................ 305 Paul Pollack On Amicable Numbers .......................................................... 321 Carl Pomerance Trigonometric Representations of Generalized Dedekind and Hardy Sums via the Discrete Fourier Transform............................. 329 Michael Th. Rassias and László Tóth On Arithmetic Properties of Products and Shifted Products ............... 345 Joël Rivat and András Sárközy Narrow Progressions in the Primes ............................................ 357 Terence Tao and Tamar Ziegler CM-Points on Straight Lines Bill Allombert, Yuri Bilu, and Amalia Pizarro-Madariaga To Helmut Maier on his 60th anniversary Abstract We prove that, with “obvious” exceptions, a CM-point .j.1/; j.2// cannot belong to a straight line in C2 defined over Q. This generalizes a result of Kühne, who proved this for the line x1 C x2 D 1. 1 Introduction In this article with or without indices denotes a quadratic1 complex number with Im>0and j denotes the j-invariant. In 1998 André [1] proved that a non-special irreducible plane curve in C2 may have only finitely many CM-points. Here a plane curve is a curve defined by an irreducible polynomial equation F.x1; x2/ D 0, where F is a polynomial with 2 complex coefficients. A CM-point in C is a point of the form .j.1/; j.2// with quadratic 1;2. Special curves are the curves of the following types: 1“Quadratic” here and below means “of degree 2 over Q ”. B. Allombert Université de Bordeaux, IMB, UMR 5251, F-33400 Talence, France CNRS, F-33400 Talence, France INRIA, F-33400 Talence, France e-mail: [email protected] Y. Bilu () Institut de Mathématiques de Bordeaux, Université de Bordeaux & CNRS, 351 cours de la Libération, 33405 Talence, France e-mail: [email protected] A. Pizarro-Madariaga Instituto de Matemáticas, Universidad de Valparaíso, Valparaíso, Chile e-mail: [email protected] © Springer International
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