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arXiv · 1809.01841

A vanishing criterion for Dirichlet series with periodic coefficients

Tapas Chatterjee·M. Ram Murty·Siddhi Pathak
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Abstract

We address the question of non-vanishing of $L(1,f)$ where $f$ is an algebraic-valued, periodic arithmetical function. We do this by characterizing algebraic-valued, periodic functions $f$ for which $L(1,f)=0$. The case of odd functions was resolved by Baker, Birch and Wirsing in 1973. We apply a result of Bass to obtain a characterization for the even functions. We also describe a theorem of the first two authors which says that it is enough to consider only the even and the odd functions in order to obtain a complete characterization.

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Tapas Chatterjee, M. Ram Murty, Siddhi Pathak. 2018-09-06. A vanishing criterion for Dirichlet series with periodic coefficients. https://arxiv.org/abs/1809.01841

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