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

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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]

• “vertical lines” x1 D j.1/; • “horizontal lines” x2 D j.2/; • modular curves Y0.N/, realized as the plane curves ˚N .x1; x2/ D 0, where ˚N is the modular polynomial of level N. Clearly, each special curve contains inﬁnitely many CM-points, and André proved that special curves are characterized by this property. André’s result was the ﬁrst non-trivial contribution to the celebrated André- Oort conjecture on the special subvarieties of Shimura varieties; see [11] and the references therein. Independently of André the same result was also obtained by Edixhoven [7], but Edixhoven had to assume the Generalized Riemann Hypothesis for certain L-series to be true. Further proof followed; we mention specially the remarkable argument of Pi- la [10]. It is based on an idea of Pila and Zannier [12] and readily extends to higher dimensions [11]. The arguments mentioned above were non-effective, because they used the Siegel-Brauer lower bound for the class number. Breuer [5] gave an effective proof, but it depended on GRH. Recently Kühne [8, 9] and, independently, Bilu et al. [3] found unconditional effective proofs of André’s theorem. Besides giving general results, both articles [9] and [3] treat also some particular curves, showing they have no CM-points at all. For instance, Kühne [9, Theorem 5] proves the following.

Theorem 1.1. The straight line x1 C x2 D 1 has no CM-points. (The same result was also independently obtained in an earlier version of [3], but did not appear in the final version.) A similar result for the curve x1x2 D 1 was obtained in [3]. One can ask about CM-points on general straight lines defined over Q; that is, defined by an equation

A1x1 C A2x2 C B D 0; (1) where A1; A2; B 2 Q. One has to exclude from consideration the special straight lines: x1 D j.1/, x2 D j.2/ and x1 D x2, the latter being nothing else than the modular curve Y0.1/ (the modular polynomial ˚1 is x1 x2). According to the theorem of André, these are the only straight lines containing inﬁnitely many CM- points. In the present paper we obtain a rather vast generalization of Theorem 1.1.

Theorem 1.2. Let .j.1/; j.2// be a CM-point belonging to a non-special straight line deﬁned over Q. Then we have one of the following options. Either

j.1/; j.2/ 2 Q; (2) CM-Points on Straight Lines 3 or

j.1/ ¤ j.2/; Q.j.1// D Q.j.2//; ŒQ.j.1// W Q D ŒQ.j.2// W Q D 2: (3)

Remark 1.3. 1. Recall that ŒQ.j./ W Q D h.O /, the class number of the “complex multiplication order” O D Endh;1i, where h;1i is the lattice generated by and 1. All orders of class number 1 and 2 are well known, which means that points satisfying (2)or(3) can be easily listed. In fact, there are 169 CM- points satisfying (2) and, up to Q-conjugacy, 217 CM-points satisfying (3); see Remark 5.3 for the details. 2. Our result is best possible because any point satisfying (2)or(3) does belong to a non-special straight line defined over Q. 3. Kühne remarks on page 5 of his article [9] that his Theorem 4 allows one, in principle, to list all possible CM-points belonging to non-special straight lines over Q, but the implied calculation does not seem to be feasible. 4. Bajolet [2] produced a software package for finding all CM-points on a given straight line. He illustrated its efficiency by proving that no straight line (1) with non-zero A1; A2; B 2 Z satisfying jA1j; jA2j; jBjÄ10 passes through a CM-point. This work is now formally obsolete because of our Theorem 1.2, but a similar method can be used in more general situations, where our theorem no longer applies. 5. CM-points .x1; x2/ satisfying x1x2 2 Q are completely classified in [4]; this generalizes the above-mentioned result from [3] about the curve x1x2 D 1. In Sects. 2 and 3 we recall basic facts about imaginary quadratic orders, class groups, ring class fields and complex multiplications. In Sect. 4 we investigate the field equality Q.j.1// D Q.j.2//. In particular, in Corollary 4.2 we determine all cases of such equality when Q.1/ ¤ Q.2/.This might be of independent interest. After all these preparations, we prove Theorem 1.2 in Sect. 5.

2 Imaginary Quadratic Orders

In this section we recall basic facts about imaginary quadratic ﬁelds and their orders, and recall a famous result of Weinberger about class groups annihilated by 2.

2.1 Class Groups

Let K be an imaginary quadratic field and O an order in K of discriminant D Df 2, where D is the discriminant of the field K (often called the fundamental discriminant) and f is the conductor of O, defined from O D Z C f OK. We denote 4 B. Allombert et al. by Cl.O/ the class group of O (the group of invertible fractional ideals modulo the invertible principal fractional ideals). As usual, we set h.O/ DjCl.O/j. Since O is uniquely determined by its discriminant , we may also write Cl./, h./,etc.In particular, Cl.D/ D Cl.OK/ is the class group of the field K, and h.D/ is the class number of K. There is a canonical exact sequence

1 ! Cl0./ ! Cl./ ! Cl.D/ ! 1; (4) where the kernel Cl0./ will be described below. This implies, in particular, that h.D/ j h./. The structure of the group Cl0./ is described, for instance, in [6], Sect. 7.D and Exercise 7.30. We brieﬂy reproduce this description here. We will assume, with a slight abuse of notation, that Z=f Z is a subring of OK=f OK. Then we have another canonical exact sequence

1 .Z= Z/.O/ , .O = O / ./ 1; ! f K f ! K f K ! Cl0 ! (5)

.O/ O .O = O / where K f is the image of the multiplicative group K in K f K . .Z= Z/.O/ .Z= Z/ The group f K f is “not much bigger” than f . Precisely, 8 ˆ <ˆ2 if D D4, f >1; Œ.Z= Z/.O/ .Z= Z/ ŒO O 3 3 >1; f K f W f D K W D ˆ if D D , f :ˆ 1 otherwise:

An easy consequence is the following formula for h./: Â Â Ã Ã fh.D/ Y D h./ D 1 p1 ; (6) ŒO W O p K pjf where .D=/ is the Kronecker symbol.

2.2 Orders with Class Groups Annihilated by 2

In this subsection we recall the famous result of Weinberger about imaginary quadratic orders whose class group is annihilated by 2. For a multiplicatively written abelian group G we denote by G2 its subgroup of squares: G2 Dfg2 W g 2 Gg. The group Cl./=Cl./2 is usually called the genus group of .Itisknownto be isomorphic to .Z=2Z/, where D ./ 2f!./ 1; !./g and !./ denote the number of distinct prime divisors. We may also remark that ./ D !./ 1 when D D (and f D 1). CM-Points on Straight Lines 5

Already Euler studied discriminants with the property

jCl./2jD1; (7)

or, equivalently, Cl./ Š .Z=2Z/. (Of course, he used a different terminology.) Chowla proved that the set of such is ﬁnite. Using a deep result of Tatuzawa [13] about Siegel’s zero, Weinberger [16] improved on this, by showing that ﬁeld discriminants D with this property are bounded explicitly with at most one exception. To state Weinberger’s result precisely, denote by D0 the square-free part of D: ( D; D Á 1 mod 4; D0 D D=4; D Á 0 mod 4:

Proposition 2.1 (Weinberger [16], Theorem 1). There exists a negative integer D such that for any discriminant D of an imaginary quadratic ﬁeld the property jCl.D/2jD1 implies jD0jÄ5460 or D D D. Using existing methods [15], it is easy to determine the full list of D with jCl.D/2jD1 and jD0jÄ5674. Since the group Cl.D/ is a quotient of the group Cl.Df 2/, if the latter has property (7) then the former does. Also, for every given D it is easy to ﬁnd all possible f such that jCl.Df 2/2jD1, using the description of the group Cl.Df 2/ given in Sect. 2.1. Hence the couples .D; f / for which jCl.Df 2/2jD1 and jD0jÄ5674 can be easily listed as well. This list is widely available in the literature since long ago; we reproduce it in Table 1. It follows that Weinberger’s result has the following consequence. Corollary 2.2. There exists a negative integer D such that jCl.Df 2/2jD1 implies that either D Df 2 appears in Table 1 or D D D.

Table 1 Known with jCl./2jD1 h./ D 1 3; 3 22; 3 32; 4;4 22; 7; 7 22; 8; 11; 19; 43; 67; 163 h./ D 2 3 42; 3 52; 3 72; 4 32; 4 42; 4 52; 7 42; 8 22; 8 32; 11 32; 15; 15 22; 20; 24; 35; 40; 51; 52; 88; 91; 115; 123; 148; 187; 232; 235; 267; 403; 427 h./ 4 3 82; 7 82; 8 62; 15 42; 15 82; 20 32; 24 22; 35 32; 40 22; 84; 88 22; 120; 120 22; 132; 168; 168 22; 195; 228; 232 22; 280; 280 22; 312; 312 22; 340; 372; 408; 408 22; 420; 435; 483; 520; 520 22; 532; 555; 595; 627; 660; 708; 715; 760; 760 22; 795; 840; 840 22; 1012; 1092; 1155; 1320; 1320 22; 1380; 1428; 1435; 1540; 1848; 1848 22; 1995; 3003; 3315; 5460 6 B. Allombert et al.

Remark 2.3. Class numbers of discriminants from Table 1 are at most 16, and the results of [15] imply that Table 1 contains all with jCl./2jD1 and h./ Ä 64. Hence if satisﬁes jCl./2jD1 but does not appear in Table 1 then we must have h./ 128. In particular, the ﬁrst two lines of Table 1 give full lists of negative quadratic discriminants with h./ D 1 and 2.

3 Ring Class Fields and Complex Multiplication

Let K be an imaginary quadratic ﬁeld, and O an order in K of discriminant D Df 2. One associates with O an abelian extension of K with Galois group Cl.O/, called the ring class ﬁeld of O. We will denote it by RiCF.O/,orRiCF./,orRiCF.K; f /. The canonical isomorphism Cl.O/ ! Gal.RiCF.O/=K/ is called the Artin map. For the details see, for instance, [6, Sect. 9]. The correspondence O $ RiCF.O/ is functorial in the following sense: if O0 is a sub-order of O, then RiCF.O0/ RiCF.O/, and we have the commutative diagram

Cl.O/ ! Gal.RiCF.O/=K/ ## Cl.O0/ ! Gal.RiCF.O0/=K/ where the horizontal arrows denote Artin maps and the vertical arrows are the natural maps of the class groups and the Galois groups. It follows that the Galois group of RiCF.O/ over RiCF.O0/ is isomorphic to the kernel of Cl.O/ ! Cl.O0/. In particular, Gal.RiCF.O/=RiCF.OK// is Cl0./, the group introduced in Sect. 2.1. (One may notice that RiCF.OK/ is nothing else but the Hilbert class ﬁeld of K.)

3.1 Compositum of Ring Class Fields

In this subsection it will be more convenient to use the “conductor notation” RiCF.K; f /. As we have seen above, if f1 j f then RiCF.K; f1/ RiCF.K; f /. It follows that the compositum of two ring class fields RiCF.K; f1/ and RiCF.K; f2/ is a subfield of RiCF.K; f /, where f D LCM.f1; f2/. It turns out that this compositum is “almost always” equal to RiCF.K; f /, but there are some exceptions. Here is the precise statement. It is certainly known, but we did not find it in the available literature.

Proposition 3.1. Let K be an imaginary quadratic ﬁeld of discriminant D and f1; f2 positive integers. Set f D LCM.f1; f2/. Then we have the following.