Dirichlet series associated to sum-of-digits functions

2018/07/20 by Everlove, Corey
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1807.07890

Abstract

We study the Dirichlet series F_b(s)=∑_n=1^∞ 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)=∑_n=1^∞ S_b(n)n^-s, where S_b(n)=∑_m=1^n-1d_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 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.

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