On special values of Dirichlet series with periodic coefficients

Let f be an algebraic valued periodic arithmetical function and L(s, f), defined as L(s, f) := Sigma(infinity)(n=1) f(n)/n(s) for R(s) > 1, be the associated Dirichlet series. In this paper, we study the vanishing and arithmetic nature of the special values L(k, f) when k > 1 is a positive integer. We prove a generalization of the Baker-Birch-Wirsing theorem conditional on the Polylog conjecture. Adopting a new approach, we define an induction operator on the space of periodic arithmetic functions, which makes precise the notion of an "imprimitive" arithmetic function. This enables us to obtain an analog of Okada's criterion for L(1, f) = 0 and derive a natural decomposition of the vector space O-k(N) = {f : Z -> Q vertical bar f(n + N) = f(n) for all n is an element of Z, L(k,f) = 0}. (C) 2021 Elsevier Inc. All rights reserved.