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

Remarks on the Selberg--Delange method