On some applications of automorphic forms to number theory

Abstract

Abstract. A basic idea of Dirichlet is to study a collection of interesting quantities {an} n≥1 by means of its Dirichlet series in a complex variable w: n≥1 ann −w . In this paper we examine this construction when the quantities an are themselves infinite series in a second complex variable s, arising from number theory or representation theory. We survey a body of recent work on such series and present a new conjecture concerning them.

Introduction: Multiple Dirichlet Series

“…See Bucur and Diaconu [19]. Nevertheless, the approach taken in [20], which we will next explain, shows that the multiple Dirichlet series cannot have meromorphic continuation to all s i and w if k > 3.…”

“…In [20] a different approach was taken. If k > 3, then it may be possible to write down a correct definition of the multiple Dirichlet series, and indeed this has essentially been done in the very interesting special case k = 4.…”

“…This yields both the meromorphic continuation and the scattering matrix, which we recall from Section 2 amounts to an action of W on Ψ such that in (22) we have Ψ ′ = σ i Ψ and more generally Z * wΨ (ws; m) = Z * Ψ (s; m). The normalizing factor in (18) works out as follows: the factor ζ α G α with α = α i is needed to normalize the Kubota Dirichlet series in (20). The other such factors are simply permuted by s −→ σ i (s).…”

“…Then, collecting terms with equal powers of |p| −s2 we have a decomposition (20) where the summation includes terms of the following type:…”

Asymptotics for sums of twisted L-functions and applications

“…Such series first arose from the study of certain Hecke-RankinSelberg type integrals of metaplectic Eisenstein series (see [BFH1]), but it is not apparent that the series used here may be so-obtained. Instead, our work is based on the convexity principle for holomorphic functions of two complex variables, whose use to study such series was first observed by Bump,Friedberg and Hoffstein [BFH2].…”