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An Isaac Newton Institute Workshop

Matrix Ensembles and L-Functions

Multiple Dirichlet series, an historical survey

Author: Dorian Goldfeld (Columbia University)

Abstract

Multiple Dirichlet series (L-functions of several complex variables) are Dirichlet series in one complex variable whose coefficients are again Dirichlet series in other complex variables. These series arise naturally in the theory of moments of zeta and L-functions. It was found recently by Diaconu-Goldfeld-Hoffstein that the moment conjectures of random matrix theory, such as the Keating-Snaith conjecture, would follow if certain multiple Dirichlet series had meromorphic continuation to a a particular tube domain.

We shall present an introduction to some of the basic definitions and techniques of this theory as well as a survey of some of the results that have been obtained by this method. These include applications to moments of L-functions, Fermat's last theorem, classification theory via Dynkin diagrams, and analysis of natural constructions as inner products of automorphic forms on GL(n).