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arXiv · 1912.03711

Zeros of Dirichlet polynomials

Arindam Roy·Akshaa Vatwani
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Abstract

We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$ and study the distribution of zeros of Dirichlet polynomials $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ corresponding to these functions. We prove that the known non-trivial zero-free half plane for Dirichlet polynomials associated to this class of multiplicative functions is optimal. We also introduce a characterization of elements in this class based on a new parameter depending on the Dirichlet series $F(s) = \sum_{n=1}^\infty f(n) n^{-s}$. In this context, we obtain non-trivial regions in which the associated Dirichlet polynomials do have zeros.

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Arindam Roy, Akshaa Vatwani. 2019-12-08. Zeros of Dirichlet polynomials. https://arxiv.org/abs/1912.03711

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