Dirichlet series of integers with missing digits

Abstract

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