On the analytic continuation of Dirichlet series with missing digits

TL;DR

This paper investigates the analytic continuation of Dirichlet series linked to integers with specific missing digits in their base-$b$ representation, providing new proofs and connections to Cantor sets and generalized Bernoulli numbers.

Contribution

It offers a new proof of the meromorphic continuation of these Dirichlet series and relates residues to a generating function inverse, connecting to Cantor sets and generalized Bernoulli numbers.

Findings

Established meromorphic continuation for these Dirichlet series.

Connected residues to a generating function inverse related to Cantor sets.

Identified residues as generalized Bernoulli numbers.

Abstract

We study the Dirichlet series associated with the integers whose radix-$b$ representation misses certain (fixed) digits. The existence of a meromorphic continuation to the entire complex plane, which was already well-known as a general fact valid for $b$-automatic Dirichlet series, is proven anew from a representation as an everywhere defined series with good convergence properties. A generating function related to the residues on the real axis is shown to be the multiplicative inverse of the moment generating function for the associated Cantor set in the unit interval. This makes the (normalized) residues some sort of generalized Bernoulli numbers.