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 3
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