# Zeros of Dirichlet Polynomials via a Density Criterion

## TL;DR

This paper establishes a criterion to determine when a half-plane contains no zeros of a Dirichlet polynomial, generalizing known results related to the Riemann hypothesis and connecting to Burnol's work.

## Contribution

It provides a necessary and sufficient density criterion for zero-free regions of Dirichlet polynomials, extending classical zero distribution results.

## Key findings

- Derived a criterion for zero-free half-planes of Dirichlet polynomials
- Generalized Be1ez-Duarte's criterion for the Riemann hypothesis
- Connected the results to Burnol's related work on Dirichlet polynomials

## Abstract

We obtain a necessary and sufficient condition in order that a semi-plane of the form $\Re(s)>r$, $r\in \mathbb{R}$, is free of zeros of a given Dirichlet polynomial. The result may be considered a natural generalization of a well-known criterion for the truth of the Riemann hypothesis due to B\'aez-Duarte. An analog for the case of Dirichlet polynomials of a result of Burnol which is closely related to B\'aez-Duarte's one is also established.