Chapter One. Type A Weyl Group Multiple Dirichlet Series

Given a root system Φ of rank r and a global field F containing the n-th roots of unity, it is possible to define a Weyl group multiple Dirichlet series whose coefficients are n-th order Gauss sums. It is a function of r complex variables, and it has meromorphic continuation to all of C, with functional equations forming a group isomorphic to the Weyl group of Φ. Weyl group multiple Dirichlet series and their residues unify many examples that have been studied previously in a case-bycase basis, often with applications to analytic number theory. (Examples may be found in the final section of the paper.) We believe these Weyl group multiple Dirichlet series are fundamental objects. The goal of this paper is to define these series for any such Φ and F , and to indicate how to study them. We will note the following points. • The coefficients of the Weyl group multiple Dirichlet series are multiplicative, but the multiplicativity is twisted , so the Dirichlet series is not an Euler product. • Due to the multiplicativity, description of the coefficients reduces to the case where the parameters are powers of a single prime p. There are only finitely many such coefficients (for given p). • In the “stable case” where n is sufficiently large (depending on Φ), the number of nonzero coefficients in the p-part is equal to the order of the Weyl group. Indeed, these nonzero coefficients are parametrized in a natural way by the Weyl group elements. • The p-part coefficient parametrized by a Weyl group element w is a product of l(w) Gauss sums, where l is the length function on the Weyl group. We note a curious similarity between this description and the coefficients of the generalized theta series on the n-fold cover of GL(n) and GL(n − 1); these coefficients are determined in Kazhdan and Patterson [17] and discussed further in Patterson [21]. See Bump and Hoffstein [11] or Hoffstein [15] for a “classical” description of these coefficients. The noted similarity means that the complete Mellin transform of the theta function would be a multiple Dirichlet series resembling our An+1 multiple Dirichlet series. There is no a priori reason that we are

A Weyl group multiple Dirichlet seriesis a Dirichlet series in several complex variables attached to a root system Φ. The number of variables equals the rank rof the root system, and the series satisfies a group of functional equations isomorphic to the Weyl group Wof Φ. In this paper we construct a Weyl group multiple Dirichlet series over the rational function field using n th order Gauss sums attached to the root system of type A 2. The basic technique is that of [11, 10]; namely, we construct a rational function in rvariables invariant under a certain action of W, and use this to build a “local factor” of the global series.

We construct multiple Dirichlet series in several complex variables whose coefficients involve quadratic residue symbols. The series are shown to have an analytic continuation and satisfy a certain group of functional equations. These are the first examples of an infinite collection of unstable Weyl group multiple Dirichlet series in greater than two variables.

To each reduced root system Φ of rank r, and each sufficiently large integer n, we define a family of multiple Dirichlet series in r complex variables, whose group of functional equations is isomorphic to the Weyl group of Φ. The coefficients in these Dirichlet series exhibit a multiplicativity that reduces the specification of the coefficients to those that are powers of a single prime p. For each p, the number of nonzero such coefficients is equal to the order of the Weyl group, and each nonzero coefficient is a product of n-th order Gauss sums. The root system plays a basic role in the combinatorics underlying the proof of the functional equations.

Constructing Affine Weyl Group Multiple Dirichlet Series

Friedberg, Hoffstein and Lieman have constructed two related multiple Dirichlet series from quadratic and higher-order L-functions and Gauss sums. We compute these multiple Dirichlet series explicitly in the case of the rational function field. This is done by utilizing the functional equation of the L-functions and the functional equation relating the two multiple Dirichlet series. We also point out a very simple correspondence between these series and their p-parts.

General multiple Dirichlet series from perverse sheaves

Multiple Dirichlet series and automorphic forms

The lower order and linear order of multiple dirichlet series

On the description of an exceptional set in Borel's relation for multiple Dirichlet series with upper restriction on the growth

The Godement-Jacquet L-function associated to the discrete subgroup SL(n, Z) (with n 2) acting on , 1972] where its analytic properties: holomorphic continuation, functional equation, Euler product, etc. were obtained by a generalization of Tate's thesis Later, [Jacquet-Piatetski-Shapiro-Shalika, 1979] obtained a different derivation of the construction of the Godement-Jacquet L-function by the use of Whittaker models.

It is shown, under some expected technical assumption, that a large class of multiple Dirichlet series which arise in the study of moments of L-functions have natural boundaries. As a remedy, we consider a new class of multiple Dirichlet series whose elements have nice properties: a functional equation and meromorphic continuation. This class suggests a notion of integral moments of L-functions.KeywordsMultiple Dirichlet seriesIntegral moments of L-functionsGood’s methodEisenstein seriesGL(3)

Dans cet article, nous considerons que certaines series de Dirichlet multiples, dont nous montrons le prolongement analytique en utilisant la formule integrale de Mellin-Barnes. Des majorations de ces series sont egalement obtenues.