Adrian Diaconu

MULTIPLE DIRICHLET SERIES AND MOMENTS OF ZETA AND L{FUNCTIONS

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.

Integral moments of automorphic L-functions

This paper exposes the underlying mechanism for obtaining second integral moments of GL2 automorphic L–functions over an arbitrary number field. Here, moments for GL2 are presented in a form enabling application of the structure of adele groups and their representation theory. To the best of our knowledge, this is the first formulation of integral moments in adele-group-theoretic terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers Q, we recover the classical results. §

Subconvexity bounds for automorphic L-functions

We break the convexity bound in the t -aspect for L -functions attached to cusp forms f for GL 2 ( k ) over arbitrary number fields k . The argument uses asymptotics with error term with a power saving, for second integral moments over spectral families of twists L ( s,f ⊗χ) by Grossencharacters χ, from our previous paper on integral moments.

A note on GL2 converse theorems

Abstract Weils well-known converse theorem shows that modular forms f∈ M k (Γ 0 (q)) are characterized by the functional equation for twists of L f ( s ). Conrey–Farmer had partial success at replacing the assumption on twists by the assumption of L f ( s ) having an Euler product of the appropriate form. In this Note we obtain a hybrid version of Weils and Conrey–Farmers results, by proving a converse theorem for all q ⩾1 under the assumption of the Euler product and, moreover, of the functional equation for the twists to a single modulus. To cite this article: A. Diaconu et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 621–624.

Moments for L-Functions for GLr×GLr-1

We establish a spectral identity for moments of Rankin–Selberg L-functions on GL r ×GL r − 1 over arbitrary number fields, generalizing our previous results for r = 2.

Natural Boundaries and Integral Moments of L-Functions

It is shown, under some expected technical assumption, that a large class of multiple Dirichlet series which arise 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. This class suggests a notion of integral moments of L-functions.