Zeros of Dirichlet polynomials

TL;DR

This paper investigates the zeros of Dirichlet polynomials linked to a specific class of multiplicative functions, establishing the optimality of known zero-free regions and characterizing the functions based on a new parameter.

Contribution

It introduces a new characterization of multiplicative functions and proves the optimality of the zero-free half-plane for their associated Dirichlet polynomials.

Findings

The zero-free half-plane for these Dirichlet polynomials is proven to be optimal.

A new parameter characterizes the class of multiplicative functions based on their Dirichlet series.

Regions where the Dirichlet polynomials have zeros are explicitly identified.

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.