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 (cid:13)c2006 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 X f(cid:18)a b(cid:19)=“main terms”+ X 1 Z ∞ σ2ir(|m|)σ2ir(|n|)F(r;m,n)dr ad−bc=1 c d m,n6=0π −∞ |nm|ir|ζ(1+2ir)|2 c>0 ∞ θ(k) ∞ +XX X ρ (m)ρ (n)F(cid:0)(cid:0)1 −k(cid:1)i;m,n(cid:1)+X X ρ (m)ρ (n)F(κ ;m,n), j,k j,k 2 j j j k=1j=1m,n6=0 j=1m,n6=0 where F(r;m,n) is a certain integral transform of f, ρ (n) denote the Fourier j coefficients for the Maass wave forms, and ρ (n) denote Fourier coefficients of j,k 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(cid:12)(cid:12)ζ∗(cid:0)1 +it,x(cid:1)(cid:12)(cid:12)2dx=logt+γ−log2π+O(cid:16)t−5/6(cid:17), 2 0 and Z 1Z 1(cid:12)(cid:12)φ∗(cid:0)x,y,1 +it(cid:1)(cid:12)(cid:12)4dxdy =2log2t+O(cid:16)(logt)5/3(cid:17), 2 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 (cid:12) (cid:12) √ (cid:12)Xn (cid:12) √ n≤ inf max (cid:12) zv(cid:12)≤ n+1 (cid:12) k(cid:12) |zi|≥1v=1,...,n2(cid:12)k=1 (cid:12) if n+1 is prime. We solve a problem posed by P. Erd˝os completely, and disprove some conjectures of P. Tur´an and 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’s book on the Riemann zeta functionandreadselectedexcerptsfrom“Reviews in number theory”. Iremember longhoursfromthatsummertryingtoprovetheLindel¨ofhypothesis,andworking on the zero density estimates. My first paper on the Hurwitz zeta function came fromthatinterest(aswellassomeideasIhadaboutBernoullipolynomialsinhigh school). Thespectraltheoryofautomorphicforms,anditsrelationshiptotheRie- 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 itcontainsomebeautifulformulae. Italsohadinterestinganddaringspeculations 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 - “AllthespectraltheoryrequiredisinBump’sSpringerlecturenotes”. Anywaythe sixthpowermomentmighthaveturnedoutmoredifficultthanoriginallythought, 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´c for 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 therecaresaboutzetafunctions; DennisHejhalforhissupportwhenIvisitedhim inMinnesota;Jan-ErikRoosforhisenthusiasmduringmyfirstyearsasagraduate 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