Non-vanishing of Dirichlet series with periodic coefficients

TL;DR

This paper investigates the non-vanishing of Dirichlet series with periodic coefficients at s=1, providing new proofs and conditions using advanced number theory techniques.

Contribution

It offers new proofs of classical non-vanishing criteria and introduces novel necessary and sufficient conditions for the non-vanishing of these Dirichlet series.

Findings

Reproved Okada's criterion for non-vanishing of L(1,f).

Extended classical results of Baker, Birch, and Wirsing.

Established new necessary and sufficient conditions for non-vanishing.

Abstract

For any periodic function $f:{\mathbb N} \to {\mathbb C}$ with period $q$, we study the Dirichlet series $L(s,f):=\sum_{n\geq 1} f(n)/n^s.$ 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 $$\rho:= q^{-1}\sum_{1\leq a\leq q} f(a).$$ 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 $L(1,f)$.