For certain sequences A of positive integers with missing g-adic digits, the Dirichlet series FA(s)=∑a∈Aa−s has abscissa convergence σc<1. The number σc is computed. This generalizes and strengthens a classical theorem Kempner on sum reciprocals sequence decimal digits.

Preface An L-function, as the term is generally understood, is a Dirichlet series in one complex variable s with an Euler product that has (at least conjecturally) analytic continuation to all complex s and a functional equation under a single reflection s → 1 − s. The coefficients are in particular multiplicative. By contrast Weyl group multiple Dirichlet series are a new class of Dirichlet se...

Weyl group multiple Dirichlet series, introduced by Brubaker, Bump, Chinta, Friedberg and Hoffstein, are expected to be Whittaker coefficients of Eisenstein series on metaplectic groups. Chinta and Gunnells constructed these multiple Dirichlet series for all the finite root systems using the method of averaging a Weyl group action on the field of rational functions. In this paper, we generalize...

The Dirichlet divisor problem is used as a model to give a conjecture concerning the conditional convergence of the Dirichlet series of an L-function.

Abstract. A Weyl group multiple Dirichlet series is a Dirichlet series in several complex variables attached to a root system Φ. The number of variables equals the rank r of the root system, and the series satisfies a group of functional equations isomorphic to the Weyl group W of Φ. In this paper we construct a Weyl group multiple Dirichlet series over the rational function field using n order...

It is shown that a large class of multiple Dirichlet series which arise naturally 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 the correct notion of integral moments of L–functions. §

As a continuation of the authors’ and Wakatsuki’s previous paper [5], we study relations among Dirichlet series whose coefficients are class numbers of binary cubic forms. We show that, for all integral models of the space of binary cubic forms, the associated Dirichlet series satisfy self dual identities.