Johan Andersson Summation formulae and zeta functions

197 Pages·2006·1.13 MB·English

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