Zeros of Dirichlet polynomials

Roy, Arindam; Vatwani, Akshaa

doi:10.1090/tran/8261

Zeros of Dirichlet polynomials
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by Arindam Roy and Akshaa Vatwani PDF

Trans. Amer. Math. Soc. 374 (2021), 643-661 Request permission

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 nontrivial 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 nontrivial regions in which the associated Dirichlet polynomials do have zeros.

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