On the abscissas of convergence of Dirichlet series

TL;DR

This paper investigates the convergence properties of a class of Dirichlet series with random exponents, providing estimates for their abscissas of convergence and absolute convergence.

Contribution

It introduces new estimates for the abscissas of convergence and absolute convergence for Dirichlet series with pairwise independent random exponents.

Findings

Established bounds for abscissas of convergence.

Derived estimates for abscissas of absolute convergence.

Analyzed the impact of independence of exponents on convergence properties.

Abstract

For the Dirichlet series of the form $\displaystyle F(z,\omega)=\sum\nolimits_{k=0}^{+\infty} f_k(\omega)e^{z\lambda_k(\omega)} $ $ (z\in\mathbb{C},$ $\omega\in\Omega)$ with pairwise independent real exponents $(\lambda_k(\omega))$ on probability space $(\Omega,\mathcal{A},P)$ an estimates of abscissas convergence and absolutely convergence are established.