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… (More)
We obtain second integral moments of automorphic L–functions on adele groups GL 2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL 1 and GL 2. This requires complete reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL 2 (Q) or… (More)
We break the convexity bound in the t–aspect for L–functions attached to cuspforms 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 [Di-Ga]. §0. Introduction In many… (More)
Let H denote the upper half-plane. A complex valued function f defined on H is called an automorphic form for Γ = SL 2 (Z), if it satisfies the following properties: (1) We have f az + b cz + d = (cz + d) κ f (z) for a b c d ∈ Γ; (2) f (iy) = O(y α) for some α, as y → ∞; (3) κ is either an even positive integer and f is holomorphic, or κ = 0, in which case,… (More)
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… (More)
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.
We exhibit a spectral identity involving L(s, Sym 2 f) for f on SL 2. Perhaps contrary to expectations, we do not treat L(s, Sym 2 f) directly as a GL 3 object. Rather, we take advantage of the coincidence that the standard L-function for SL 2 is the symmetric square for a cuspform on GL 2 restricted to SL 2.  As SL 2 = Sp 2 , the integral identities… (More)
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… (More)