Multiple Dirichlet Series and Moments of Zeta and L–functions
نویسندگان
چکیده
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 central 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. §
منابع مشابه
Relative order and type of entire functions represented by Banach valued Dirichlet series in two variables
In this paper, we introduce the idea of relative order and type of entire functions represented by Banach valued Dirichlet series of two complex variables to generalize some earlier results. Proving some preliminary theorems on the relative order, we obtain sum and product theorems and we show that the relative order of an entire function represented by Dirichlet series is the same as that of i...
متن کاملq-BERNOULLI NUMBERS AND POLYNOMIALS ASSOCIATED WITH MULTIPLE q-ZETA FUNCTIONS AND BASIC L-SERIES
By using q-Volkenborn integration and uniform differentiable on Zp, we construct p-adic q-zeta functions. These functions interpolate the qBernoulli numbers and polynomials. The value of p-adic q-zeta functions at negative integers are given explicitly. We also define new generating functions of q-Bernoulli numbers and polynomials. By using these functions, we prove analytic continuation of som...
متن کاملNatural Boundaries and a Correct Notion of Integral Moments of L–functions
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. This class suggests the correct notion of integral moments of L–functions. §
متن کاملNatural Boundaries and the Correct Notion of Integral Moments of L–functions
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. §
متن کاملGeneralized Ritt type and generalized Ritt weak type connected growth properties of entire functions represented by vector valued Dirichlet series
In this paper, we introduce the idea of generalized Ritt type and generalised Ritt weak type of entire functions represented by a vector valued Dirichlet series. Hence, we study some growth properties of two entire functions represented by a vector valued Dirichlet series on the basis of generalized Ritt type and generalised Ritt weak type.
متن کامل