Remarks on the Selberg–Delange method

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