Dirichlet series of integers with missing digits

TL;DR

This paper analyzes the Dirichlet series of integer sequences with missing digits, computing their abscissa of convergence, and generalizes Kempner's classical theorem on missing decimal digit sequences.

Contribution

It extends classical results by explicitly calculating the abscissa of convergence for sequences with missing digits, broadening understanding of their Dirichlet series.

Findings

Computed the abscissa of convergence for sequences with missing g-adic digits.

Generalized Kempner's theorem to broader classes of missing digit sequences.

Established conditions under which the Dirichlet series converges for these sequences.

Abstract

For certain sequences $A$ of positive integers with missing $g$-adic digits, the Dirichlet series $F_A(s) = \sum_{a\in A} a^{-s}$ has abscissa of convergence $\sigma_c < 1$. The number $\sigma_c$ 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.