Sums of L-functions over rational function fields

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

General multiple Dirichlet series from perverse sheaves

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.

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.

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.

Multiple Dirichlet series and automorphic forms

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)

The lower order and linear order of multiple dirichlet series

In 1961, Chandrasekharan and Narasimhan showed that for a large class of Dirichlet series the functional equation and two types of arithmetical identities are equivalent. In 1992, Hawkins and Knopp proved a Hecke correspondence theorem for modular integrals with rational period function on theta group. Analogous to Chandrasekharan and Narasimhan, in 2015 Sister Ann M. Heath has shown that the functional equation in Hawkins and Knopp context and two type of arithmetical identities are equivalent. She considered the functional equation and showed its equivalence to two arithmetical identities associated with entire modular cusp integrals involving rational period functions for the full modular group. In this paper we extend the results of Sister Ann M. Heath to entire automorphic integrals involving rational period functions on discrete Hecke group. Keywords Recurrence; Bernoulli numbers; Bernoulli polynomials; Hypergeometric Bernoulli numbers; Polynomials

A Hecke correspondence theorem for automorphic integrals with symmetric rational period functions on the Hecke groups

In this note we treat the functional equation \(f(cz)=a(z)f(z)+b(z)\), where c is a constant \(|c|\ne1\), 0, and a(z), b(z) are rational functions. It is shown that no transcendental meromorphic solution of the functional equation satisfies an algebraic differential equation with rational coefficients.

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We present a well-parallelizable algorithm which, taking a straight-line program for the evaluation of a vectorial field of rational functions of Q(X 1,...,X n ) as input, decides whether they allow a rational potential function and, in case of affirmative answer, computes it as output. We introduce a mixed model of representation of polynomials to allow the application of integration techniques and show how to perform some basic operations with it. The algorithm is presented as a family of arithmetic networks of polynomial size and poly logarithmic depth in the degree of the occurring polynomials.KeywordsRational FunctionPartial FractionGreat Common DivisorProbabilistic AlgorithmParallel TimeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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

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.