Dirichlet series associated to sum-of-digits functions

TL;DR

This paper investigates the analytic properties of Dirichlet series linked to sum-of-digits functions in various bases, revealing their meromorphic continuations, pole structures, and explicit residues.

Contribution

It provides the first detailed analysis of the meromorphic continuation and pole structure of Dirichlet series associated with sum-of-digits functions in arbitrary bases.

Findings

Dirichlet series extend meromorphically to the complex plane.

Explicit formulas for residues at poles are derived.

Continuous interpolation of sum-of-digits functions to non-integer bases is established.

Abstract

We study the Dirichlet series $F_b(s)=\sum_{n=1}^\infty d_b(n)n^{-s}$, where $d_b(n)$ is the sum of the base-$b$ digits of the integer $n$, and $G_b(s)=\sum_{n=1}^\infty S_b(n)n^{-s}$, where $S_b(n)=\sum_{m=1}^{n-1}d_b(m)$ is the summatory function of $d_b(n)$. We show that $F_b(s)$ and $G_b(s)$ have continuations to the plane $\mathbb{C}$ as meromorphic functions of order at least 2, determine the locations of all poles, and give explicit formulas for the residues at the poles. We give a continuous interpolation of the sum-of-digits functions $d_b$ and $S_b$ to non-integer bases using a formula of Delange, and show that the associated Dirichlet series have a meromorphic continuation at least one unit left of their abscissa of absolute convergence.