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...