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.

Abstract. This paper exposes the underlying mechanism for obtaining second integral moments of GL 2 automorphic L-functions over an arbitrary number field. Here, moments for GL 2 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.

We establish the meromorphic continuation of a multiple Dirichlet series associated to the fourth moment of quadratic Dirichlet L-functions, over the rational function field Fq(T) with q odd, up to its natural boundary. This is the first such result in which the group of functional equations is infinite; in such cases, it is expected that the series cannot be continued everywhere but can at least be extended to a large enough region to deduce asymptotics at the central point. In this case, these asymptotics coincide with existing predictions for the fourth moment of the symplectic family of quadratic Dirichlet Lfunctions. The construction uses the Weyl group action of a particular Kac-Moody algebra; this suggests an approach to higher moments using appropriate non-affine Kac-Moody algebras.

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We break the convexity bound in thet-aspect forL-functions attached to cusp formsffor GL2(k) over arbitrary number fieldsk. The argument uses asymptotics with error term with a power saving, for second integral moments over spectral families of twistsL(s,f⊗χ) by Grossencharacters χ, from our previous paper on integral moments.