Zeros of Dirichlet polynomials

Abstract

We consider a certain class of multiplicative functions f : N → C and study the distribution of zeros of Dirichlet polynomials FN (s) = n≤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 seriesIn this context, we obtain non-trivial regions in which the associated Dirichlet polynomials do have zeros.