On the Number of Zeros and Poles of Dirichlet Series

TL;DR

This paper establishes lower bounds on the zeros and poles of Dirichlet series, solving an open problem, and explores applications to Picard theorems, zero distribution, and uniqueness issues.

Contribution

It provides new lower bounds on zeros and poles of Dirichlet series and addresses an open problem by Bombieri and Perelli.

Findings

Confirmed an affirmative bound for zeros and poles

Applied results to Picard theorems and zero distribution

Enhanced understanding of uniqueness in Dirichlet series

Abstract

This paper investigates lower bounds on the number of zeros and poles of a general Dirichlet series in a disk of radius $r$ and gives, as a consequence, an affirmative answer to an open problem of Bombieri and Perelli on the bound. Applications will also be given to Picard type theorems, global estimates on the symmetric difference of zeros, and uniqueness problems for Dirichlet series.