Abstract
For a rational valued periodic function, we associate a Dirichlet series and provide a new necessary and sufficient condition for the vanishing of this Dirichlet series specialized at positive integers. This question was initiated by Chowla and carried out by Okada for a particular infinite sum. Our approach relies on the decomposition of the Dirichlet characters in terms of primitive characters. Using this, we find some new family of natural numbers for which a conjecture of Erdős holds. We also characterize rational valued periodic functions for which the associated Dirichlet series vanishes at two different positive integers under some additional conditions.
Citation
Abhishek Bharadwaj. "On Primitivity and Vanishing of Dirichlet Series." Michigan Math. J. Advance Publication 1 - 35, 2024. https://doi.org/10.1307/mmj/20226269
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