Proceedings of Symposia in Pure Mathematics

Volume 75, 2006

M u l t i p l e Dirichle t Serie s a n d A u t o m o r p h i c F o r m s

Gautam Chinta, Solomon Friedberg, and Jeffrey Hoffstein

ABSTRACT. This article gives an introduction to the multiple Dirichlet series

arising from sums of twisted automorphic L-functions. We begin by explaining

how such series arise from Rankin-Selberg constructions. Then more recent

work, using Hartogs' continuation principle as extended by Bochner in place

of such constructions, is described. Applications to the nonvanishing of L-

functions and to other problems are also discussed, and a multiple Dirichlet

series over a function field is computed in detail.

1. Motivation

Of the major open problems in modern mathematics, the Riemann hypothesis,

which states that the nontrivial zeroes of the Riemann zeta function ((s) lie on the

line 5R(s) = | , is one of the deepest and most profoundly important. A consequence

of the Riemann Hypothesis which has far reaching applications is the Lindelof

Hypothesis. This states that for any e 0 there exists a constant C(e) such that

for all t,

|C(l/2 + it)|C(e)|*|

e

.

The Lindelof Hypothesis remains as unreachable today as it was 100 years ago, but

there has been a great deal of progress in obtaining approximations of it. These

are results of the form |C(l/2 + it)\ C(e)\t\K,+€1 where K 0 is some fixed real

number. For example, Riemann's functional equation for the zeta function, together

with Stirling's approximation for the gamma function and the Phragmen-Lindelof

principle, are sufficient to obtain what is known as the convexity bound for the zeta

function, namely K = | , or: |£ ( | + it) | C(e)|£|4+e.

Any improvement over \ in this upper bound is known as "breaking convexity."

There are also many known generalizations of £(s) and analogous definitions of

convexity breaking that are viewed with great interest. This is, first, because of the

1991 Mathematics Subject Classification. Primary 11-02, 11F66, 11M41; Secondary 11F37,

11F70, 11M06.

Key words and phrases, multiple Dirichlet series, automorphic form, twisted L-function,

mean value of L-functions, Gauss sum.

The first author was supported in part by NSF Grant DMS-0354534 and a grant from the

Reidler Foundation.

The second author was supported in part by NSF Grant DMS-0353964.

The third author was supported in part by NSF Grant DMS-0354534.

©2006 American Mathematical Society

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http://dx.doi.org/10.1090/pspum/075/2279929