Abstract
We study the Dirichlet series [Formula: see text], where [Formula: see text] is the sum of the base-b digits of the integer n, 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.
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