Non-vanishing of Dirichlet series with periodic coefficients

Abstract

For any periodic function $f:\mathbb{N}\to \mathbb{C}$ with period $q$, we study the Dirichlet series $L(s,f):={\sum }_{n\ge 1}f(n)/n^s.$ 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 residue $$\rho :=q^{-1}\sum _{1\le a\le q}f(a).$$ Thus, the function is analytic at $s=1$ when $\rho =0$ 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 $L(1,f)$.

22 pages