Compositio Mathematica



Multiple Dirichlet Series and Moments of Zeta and L-Functions


Adrian Diaconu a1, Dorian Goldfeld a2 and Jeffrey Hoffstein a3
a1 Department of Mathematics, Columbia University, New York, NY 10027, U.S.A. e-mail: cad@math.columbia.edu
a2 Department of Mathematics, Columbia University, New York, NY 10027, U.S.A. e-mail: goldfeld@math.columbia.edu
a3 Department of Mathematics, Brown University, Providence, RI 02912, U.S.A. e-mail: jhoff@math.brown.edu

Article author query
diaconu a   [Google Scholar] 
goldfeld d   [Google Scholar] 
hoffstein j   [Google Scholar] 
 

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.


Key Words: L-functions; moments; multiple Dirichlet series; twists; zeta functions.