A vanishing criterion for dirichlet series with periodic coefficients

Abstract

We address the question of non-vanishing of L(1, f) where f is an algebraic- valued, periodic arithmetical function. We do this by characterizing algebraic-valued, periodic functions f for which L(1, f) = 0. The case of odd functions was resolved by Baker, Birch and Wirsing in 1973. We apply a result of Bass to obtain a charac- terization for the even functions. We also describe a theorem of the first two authors which says that it is enough to consider only the even and the odd functions in order to obtain a complete characterization.