This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as a consequence, precise asymptotics (previously conjectured via random matrix theory) for moments o...

For the random Dirichlet series ∞ ∑ n=0 Xn(ω) e−sλn (s = σ + it ∈ C, 0 = λ0 < λn ↑ ∞), whose coefficients are uniformly nondegenerate independent random variables, we provide some explicit conditions for the line of convergence to be its natural boundary a.s. Running Title Natural Boundary of Random Dirichlet Series

It is shown that a large class of multiple Dirichlet series which arise naturally in the study of moments of L–functions have natural boundaries. As a remedy we consider a new class of multiple Dirichlet series whose elements have nice properties: a functional equation and meromorphic continuation. We believe this class reveals the correct notion of integral moments of L–functions. §

For an even Dirichlet character , we obtain a formula for L(1;) in terms of a sum of Dirichlet L-series evaluated at s = 2 and s = 3 and a rapidly convergent numerical series involving the central binomial coeecients. We then derive a class number formula for real quadratic number elds by taking L(s;) to be the quadratic L-series associated with these elds.

Weyl group multiple Dirichlet series were associated with a root system Φ and a number field F containing the n-th roots of unity by Brubaker, Bump, Chinta, Friedberg and Hoffstein [3] and Brubaker, Bump and Friedberg [4] provided n is sufficiently large; their coefficients involve n-th order Gauss sums and reflect the combinatorics of the root system. Conjecturally, these functions coincide wi...

The main aim of this paper is to obtain a converse theorem for double Dirichlet series and use it to show that the Shintani zeta functions [13] which arise in the theory of prehomogeneous vector spaces are actually linear combinations of Mellin transforms of metaplectic Eisenstein series on GL(2). The converse theorem we prove will apply to a very general family of double Dirichlet series which...

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...

This article gives an introduction to the multiple Dirichlet series arising from sums of twisted automorphic L-functions. We begin by explaining how such series arise from Rankin-Selberg constructions. Then more recent work, using Hartogs’ continuation principle as extended by Bochner in place of such constructions, is described. Applications to the nonvanishing of Lfunctions and to other probl...

A Weyl group multiple Dirichlet series is a Dirichlet series in several complex variables attached to a root system Φ. The number of variables equals the rank r of the root system, and the series satisfies a group of functional equations isomorphic to the Weyl group W of Φ. In this paper we construct a Weyl group multiple Dirichlet series over the rational function field using n order Gauss sum...