Convergence Abscissas for Dirichlet Series with Multiplicative Coefficients

TL;DR

This paper investigates the relationships between different convergence abscissas of Dirichlet series with multiplicative coefficients, clarifying how these abscissas compare in such cases.

Contribution

It provides new insights into the convergence properties of Dirichlet series with multiplicative coefficients, specifically relating simple, uniform, and absolute convergence abscissas.

Findings

Established bounds between convergence abscissas for multiplicative coefficients

Extended classical results to the case of completely multiplicative coefficients

Clarified the differences in convergence behavior based on coefficient multiplicativity

Abstract

This note deals with the relationship between the abscissas of simple, uniform and absolute convergence for the Dirichlet series $f(s) = \sum_{n=1}^\infty a_n n^{-s}$, when the coefficients $a_n$ are either multiplicative or completely multiplicative.