Non-vanishing of Dirichlet series with periodic coefficients

Abstract

For any periodic function f:N→C with period q, we study the Dirichlet series L(s,f):=∑n≥1f(n)/ns. It is well-known that this admits an analytic continuation to the entire complex plane except at s=1, where it has a simple pole with residueρ:=q−1∑1≤a≤qf(a). Thus, the function is analytic at s=1 when ρ=0 and in this case, we study its non-vanishing using the theory of linear forms in logarithms and Dirichlet L-series. In this way, we give new proofs of an old criterion of Okada for the non-vanishing of L(1,f) as well as a classical theorem of Baker, Birch and Wirsing. We also give some new necessary and sufficient conditions for the non-vanishing of L(1,f).