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Johan Andersson Summation Formulae and Zeta Functions

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Johan Andersson Summation Formulae and Zeta Functions Johan Andersson Summation formulae and zeta functions Department of Mathematics Stockholm University 2006 Johan Andersson Summation formulae and zeta functions 4 Doctoral Dissertation 2006 Department of mathematics Stockholm University SE-106 91 Stockholm Typeset by LATEX2e c 2006 by Johan Andersson e-mail: [email protected] ISBN 91-7155-284-7 Printed by US-AB, Stockholm, 2006. ABSTRACT In this thesis we develop the summation formula Z ∞ X a b X 1 σ2ir(|m|)σ2ir(|n|)F (r; m, n)dr f = “main terms”+ c d π ir 2 ad−bc=1 m,n6=0 −∞ |nm| |ζ(1 + 2ir)| c>0 ∞ θ(k) ∞ X X X 1 X X + ρj,k(m)ρj,k(n)F 2 − k i; m, n + ρj(m)ρj(n)F (κj; m, n), k=1 j=1 m,n6=0 j=1 m,n6=0 where F (r; m, n) is a certain integral transform of f, ρj(n) denote the Fourier coefficients for the Maass wave forms, and ρj,k(n) denote Fourier coefficients of holomorphic cusp forms of weight k. We then give some generalisations and ap- plications. We see how the Selberg trace formula and the Eichler-Selberg trace formula can be deduced. We prove some results on moments of the Hurwitz and Lerch zeta-function, such as Z 1 ∗ 1 2 −5/6 ζ 2 + it, x dx = log t + γ − log 2π + O t , 0 and Z 1 Z 1 ∗ 1 4 2 5/3 φ x, y, 2 + it dxdy = 2 log t + O (log t) , 0 0 where ζ∗(s, x) and φ∗(x, y, s) are modified versions of the Hurwitz and Lerch zeta functions that make the integrals convergent. We also prove some power sum results. An example of an inequality we prove is that n √ √ X v n ≤ inf max zk ≤ n + 1 |z |≥1 v=1,...,n2 i k=1 if n + 1 is prime. We solve a problem posed by P. Erd˝oscompletely, and disprove some conjectures of P. Tur´anand K. Ramachandra. PREFACE I have been interested in number theory ever since high school when I read clas- sics such as Hardy-Wright. My special interest in the zeta function started in the summer of 1990 when I studied Aleksandar Ivi´c’sbook on the Riemann zeta function and read selected excerpts from “Reviews in number theory”. I remember long hours from that summer trying to prove the Lindel¨ofhypothesis, and working on the zero density estimates. My first paper on the Hurwitz zeta function came from that interest (as well as some ideas I had about Bernoulli polynomials in high school). The spectral theory of automorphic forms, and its relationship to the Rie- mann zeta function has been an interest of mine since 1994, when I first visited Matti Jutila in Turku, and he showed a remarkable paper of Y¯oichi Motohashi to me; “The Riemann zeta function and the non-euclidean Laplacian”. Not only did it contain some beautiful formulae. It also had interesting and daring speculations on what the situation should be like for higher power moments. The next year I visited a conference in Cardiff, where Motohashi gave a highly enjoyable talk on the sixth power moment of the Riemann zeta function. One thing I remember - “All the spectral theory required is in Bump’s Springer lecture notes”. Anyway the sixth power moment might have turned out more difficult than originally thought, but it further sparked my interest, and I daringly dived into Motohashi’s inter- esting, but highly technical Acta paper. My first real break-through in this area came in 1999, when I discovered a new summation formula for the full modular group. Even if I have decided not to include much related to Motohashi’s original theory (the fourth power moment), the papers 7-13 are highly influenced by his work. Although somewhat delayed, I am pleased to finally present my results in this thesis. Johan Andersson ACKNOWLEDGMENTS First I would like to thank the zeta function troika Aleksandar Ivi´c, Y¯oichi Mo- tohashi and Matti Jutila. Aleksandar Ivi´cfor his valuable comments on early versions of this thesis, as well as his text book which introduced me to the zeta function; Y¯oichi Motohashi for his work on the Riemann zeta function which has been a lot of inspiration, and for inviting me to conferences in Kyoto and Ober- wolfach; Matti Jutila for first introducing me to Motohashi’s work, and inviting me to conferences in Turku. Further I would like to thank Kohji Matsumoto and Masanori Katsurada, for being the first to express interest in my first work, and showing that someone out there cares about zeta functions; Dennis Hejhal for his support when I visited him in Minnesota; Jan-Erik Roos for his enthusiasm during my first years as a graduate student; My friends at the math department (Anders Olofsson, Jan Snellman + others) for mathematical discussions and a lot of good times; My family: my parents, my sister and especially Winnie and Kevin for giving meaning to life outside of the math department. And finally I would like to thank my advisor Mikael Passare, for his never ending patience, and for always believing in me. Johan Andersson CONTENTS Introduction ................................... 13 Part I - The power sum method 21 Introduction ................................... 23 On some Power sum problems of Tur´anand Erd˝os .............. 29 Disproof of some conjectures of P Tur´an .................... 41 Disproof of some conjectures of K Ramachandra ............... 47 Part II - Moments of the Hurwitz and Lerch zeta functions 53 Introduction ................................... 55 Mean value properties of the Hurwitz zeta function .............. 61 On the fourth power moment of the Hurwitz zeta-function .......... 67 On the fourth power moment of the Lerch zeta-function ........... 79 Part III - A new summation formula on the full modular group and Klooster- man sums 89 Introduction ................................... 91 A note on some Kloosterman sum identities .................. 99 A note on some Kloosterman sum identities II ................ 107 A summation formula on the full modular group ............... 115 Contents 12 A summation formula over integer matrices .................. 133 A summation formula over integer matrices II ................. 145 The summation formula on the modular group implies the Kuznetsov summa- tion formula .................................. 159 The Selberg and Eichler-Selberg trace formulae ................ 169 INTRODUCTION This thesis in analytic number theory consists of three parts and altogether thirteen papers. Each of the parts has a separate introduction. This thesis is by no means complete and is a result of a number compromises. There are additional papers I would have liked to include, some of which I refer to as forthcoming papers. Even if they are mostly complete they would still need some more work to be included in this thesis. However, even if they would have painted a fuller picture and provided further motivation of my main result, the current version does not depend upon them and it is entirely self contained. Part I and parts of part II of this thesis have also been included in my licentiate thesis [3], and three of the papers have been published in mathematical journals ([1], [2] and [4]). The Riemann zeta-function The Riemann zeta-function1 ∞ X ζ(s) = n−s (Re(s) > 1) (0.1) n=1 is perhaps the deepest function in all of mathematics. First of all it is arithmetic Y 1 ζ(s) = , (Re(s) > 1) (0.2) 1 − p−s p prime and has an Euler product. It is symmetric π ζ(s) = χ(s)ζ(1 − s) χ(s) = 2sπs−1 sin s Γ(1 − s) (0.3) 2 and has a functional equation. Furthermore it is random. A theorem illustrating this is the Voronin universality theorem (see e.g [8]). If f is a non-vanishing analytic function on D for a compact subset D ⊂ {s ∈ C : 1/2 < Re(s) < 1}, there exists a sequence Tk = Tk(D, f) so that lim max |ζ(s + iTk) − f(s)| = 0. (0.4) k→∞ s∈D 1 Introduced by Euler [6] and studied by Riemann [10]. General references: Ivi´c[7] and Titchmarsh [12]. Introduction 14 The depth of the function comes from its simple definition and these three prop- erties. I particular The Euler product implies that the function contains all infor- mation about the primes, and therefore its study belongs in the realms of number theory. The most fundamental open problem about the Riemann zeta-function is the Riemann hypothesis2 that says that ζ(s) = 0 implies that either Re(s) = 1/2 or s is an even negative integer. Of near equal importance are the weaker conjectures 1. The Lindel¨ofhypothesis: |ζ(1/2 + it)| = O(|t|), ∀ > 0. 2. The density hypothesis: N(σ, T ) = O(T 2(1−σ)+), ∀ > 0, where N(σ, T ) denote the number of zeroes ρ with real part Re(ρ) ≥ σ, and imagi- nary part −T ≤ Im(ρ) ≤ T . It is obvious that the Riemann hypothesis implies the density hypothesis and with some complex analysis, it is possible to show that the Riemann hypothesis implies the Lindel¨ofhypothesis, and the Lindel¨ofhypothesis implies the density hypothesis. Many consequences of the Riemann hypothesis are in fact implied already by the density hypothesis. Part I - The power sum method In an attempt to prove the density hypothesis we were lead to study the Tur´an power sum method. The method allows us to obtain lower bounds for n X v max 1 + zk , v=M(n),...,N(n) k=1 and in the first papers on his method Tur´anstated some conjectures on these bounds that would indeed imply the density hypothesis.
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