Zeros of Dirichlet polynomials

Abstract

We consider a certain class of multiplicative functions $f:\mathbb{N}\to \mathbb{C}$ 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.

15 pages, 1 figure