Some remarks on the convergence of the Dirichlet series of L-functions and related questions

TL;DR

This paper investigates the convergence properties of Dirichlet series associated with L-functions in the Selberg class, revealing differences within extended classes and exploring implications for zeros and independence.

Contribution

It proves the coincidence of abscissae of uniform and absolute convergence for Selberg class L-functions and analyzes their behavior in extended classes, linking to zeros and independence.

Findings

Abscissae of uniform and absolute convergence coincide for Selberg class L-functions.

Different convergence behavior observed in extended Selberg class.

Links established between majorants, zero distribution, and independence results.

Abstract

First we show that the abscissae of uniform and absolute convergence of Dirichlet series coincide in the case of $L$-functions from the Selberg class $\mathcal{S}$. We also study the latter abscissa inside the extended Selberg class, indicating a different behavior in the two classes. Next we address two questions about majorants of functions in $\mathcal{S}$, showing links with the distribution of the zeros and with independence results.