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arXiv · 1405.6982

Non-vanishing of Dirichlet series with periodic coefficients

Tapas Chatterjee·M. Ram Murty
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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 $L(1,f)$.

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Tapas Chatterjee, M. Ram Murty. 2014-05-27. Non-vanishing of Dirichlet series with periodic coefficients. https://arxiv.org/abs/1405.6982

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