Soit $p_k$ le $k$-ième nombre premier. Une fonction arithmétique complètement additive est définie sur ${\mathbb{N}}^{*}$ par la donnée des $f\left(p_k\right)$ et la formule $f\left(n\right)={\sum }_{k\ge 1}f(p_k)v_{p_k}\left(n\right)$$(n\ge 1)$, où $v_p$ désigne la...

Let Ω(n) and ω(n) denote the number of distinct prime factors of the positive integer n, counted respectively with and without multiplicity. Let $d_k\left(n\right)$ denote the Piltz function (which counts the number of ways of writing n as a product of k factors). We obtain a precise estimate of the sum ${\sum }_{n\le x,\Omega \left(n\right)-\omega \left(n\right)=q}f\left(n\right)$ for a class of multiplicative functions f, including in particular $f\left(n\right)=d_k\left(n\right)$, unconditionally if 1 ≤ k ≤ 3, and under some reasonable assumptions if k ≥ 4. The result also applies to f(n) = φ(n)/n (where φ is the totient...