Remarks on the Selberg--Delange method

TL;DR

This paper explores alternative hypotheses to the classical Selberg-Delange method, providing new conditions under which sharp asymptotic estimates for averages of multiplicative functions can be obtained.

Contribution

It introduces different, non-traditionally comparable hypotheses that still allow for precise asymptotic analysis of multiplicative functions with specific Dirichlet series forms.

Findings

New hypotheses enable sharp asymptotic estimates

Broader conditions than classical assumptions

Enhanced understanding of multiplicative function averages

Abstract

Let $\varrho$ be a complex number and let $f$ be a multiplicative arithmetic function whose Dirichlet series takes the form $\zeta(s)^\varrho G(s)$, where $G$ is associated to a multiplicative function $g$. The classical Selberg-Delange method furnishes asymptotic estimates for averages of $f$ under assumptions of either analytic continuation for $G$, or absolute convergence of a finite number of derivatives of $G(s)$ at $s=1$. We consider different set of hypotheses, not directly comparable to the previous ones, and investigate how they can yield sharp asymptotic estimates for the averages of~$f$.