MULTIPLE DIRICHLET SERIES AND AUTOMORPHIC FORMS 5

for d in this range and, thus, the Lindelof Hypothesis in the d aspect is true "on

average." Results of this type are significant in their own right and can also have

important applications.

One of the major breakthroughs in analytic number theory in the last 5 years

has been the following discovery: The assumption that the zeros of L-functions

are distributed in the same way as the eigenvalues of random hermitian matrices

allows one to obtain precise conjectures on the statistical distribution of values of

L-functions. For example, the conjectured moments of the Riemann zeta function,

by Keating and Snaith [43], were unattainable until the incorporation of random

matrix models into the theory. A major connection between this work and multi-

ple Dirichlet series was observed in [27] where it was shown that the conjectures

obtained by random matrix theory could also be read off from the polar divisors of

certain multiple Dirichlet series. It seems likely that multiple Dirichlet series will

play a key role in the future study of the statistical distribution of L-values.

In this article we discuss generalizations of the function Z(s,w) introduced

above, generalizations that capture the behavior of a family of twists of an auto-

morphic L-function. We describe different methods for obtaining the meromorphic

continuations of such objects, and consequences that can be drawn from the con-

tinuations. Section 2 introduces the families of twisted L-functions of concern. It

also describes a number of Rankin-Selberg constructions that give rise to double

Dirichlet series. Section 3 concerns quadratic twists. We begin with a heuristic

that explains why these sums of twisted L-functions should have continuation in w

beyond the region of absolute convergence. We next describe the several-complex-

variable methods that seem most effective in terms of continuation of the multiple

Dirichlet series. We conclude with various applications, of interest both in their

own right and also as illustrations of the kinds of theorems that can be established

by these methods. Section 4 concerns higher order twists. The situation concern-

ing sums of higher twists is more complicated, with Gauss sums playing a key role,

and in the few known examples one is led to continue several different families of

weighted series simultaneously. Once again, various applications are presented. Sec-

tion 5 gives an explicit example in the function field setting, where many multiple

Dirichlet series can be shown to be rational functions in several complex variables.

The final section gives some additional examples and concluding remarks.

2. The Family of Twists of a Given L-Function

2.1. The basic questions. Fix an integer n 2 and let F be a global field

containing all n-th roots of unity. (The reader may choose to focus on number

fields now, but in Section 5 we will give a concrete example in the function field

case.) Let T T be a fixed automorphic representation of GL(r) over the field F, with

standard L-function

L(s,7r) = y^c(m)|m|~ s

for Sft(s) sufficiently large. (In this article L(S,TT) refers to the finite part of the L-

function.) Here \m\ denotes (an abuse of notation) an absolute norm. Throughout

the paper we normalize all L-functions to have functional equation under s \-Â± \ â€” s.

Then our basic problem is to study the family of twisted L-functions

L(s,rr x x) = ^ , c ( m ) x ( m ) | m p

s