Zeros of Dirichlet polynomials

We consider a certain class of multiplicative functions f : ℕ → C and study the distribution of zeros of Dirichlet polynomials FN(s) = Σn≤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) = Σ∞n=1 f(n)n-s. In this context, we obtain nontrivial regions in which the associated Dirichlet polynomials do have zeros.