Affiliation:
1. Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043, USA
Abstract
We study the Dirichlet series [Formula: see text], where [Formula: see text] is the sum of the base-[Formula: see text] digits of the integer [Formula: see text], and [Formula: see text], where [Formula: see text] is the summatory function of [Formula: see text]. We show that [Formula: see text] and [Formula: see text] have analytic continuations to the plane [Formula: see text] 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 [Formula: see text] and [Formula: see text] 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.
Funder
Directorate for Mathematical and Physical Sciences
Publisher
World Scientific Pub Co Pte Lt
Subject
Algebra and Number Theory