Multiple Dirichlet Series and Moments of Zeta and L-Functions

Abstract

Abstract. 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 of zeta functions and quadratic L-series. As an application of the theory, in a third section, we obtain the current best known error term for mean values of cubes of cent ral values of Dirichlet L-series. The methods utilized to derive this result are the convexity principle for functions of several complex-variables combined with a knowledge of groups of functional equations for certain multiple Dirichlet series.

Mean values of biquadratic zeta functions

“…If this can be done, the multiple Dirichlet series is said to be "perfect." This procedure has been carried out in detail for the Dynkin diagrams A 2 , A 3 and D 4 , see [14], [9], [6], [5], [4]. These multiple Dirichlet series give asymptotics for mean values of quadratic twists of L-functions on GL(1), GL(2) and GL(3).…”

Introduction: Multiple Dirichlet Series

“…Taking t = 0, the only terms that contribute will be those in the image of a map B λ −→ B λ+ρ , and the sum reduces to a sum over B λ . Thus when t = 0, the formula reduces to (32). Most importantly for us, taking t = q −1 and comparing with (29), we see that the sum over B λ+ρ is exactly δ 1/2 (a λ )W • (a λ ).…”

“…It is possible regard (4) as a multiple Dirichlet series, and indeed both (3) and (4) are treated together in Diaconu, Goldfeld and Hoffstein [32]. See [31] in this volume for a discussion of the sixth integral moment and its connection with the spectral theory of Eisenstein series on GL 3 .…”

“…Only three of them, for Cartan types A 2 , A 3 and D 4 , are related to moments of L-functions. The case k = 3, related to D 4 , was applied in [32] to the third moment after the combinatorics needed to precisely define the Dirichlet series were established in [21].…”

Asymptotics for sums of twisted L-functions and applications

“…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]. In [DGH, a similar point of view was taken. In particular, in [DGH] the approach of searching for a multiple Dirichlet series which is a weighted sum of twisted L-functions with a full meromorphic continuation in all variables was formalized.…”