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.
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)|*|
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,
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
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