Remarks on the Selberg–Delange method
Volume 200 / 2021
Acta Arithmetica 200 (2021), 349-369
MSC: Primary 11N37.
DOI: 10.4064/aa201024-26-5
Published online: 27 September 2021
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 the averages of $f$ under the assumptions of either analytic continuation for $G$, or absolute convergence of a finite number of derivatives of $G(s)$ at $s=1$. We consider a different set of hypotheses, not directly comparable to the previous ones, and investigate how they can yield sharp asymptotic estimates for the averages of $f$.
