Sur un problème de Rényi et Ivić concernant les fonctions de diviseurs de Piltz
Volume 161 / 2013
Abstract
Let $\varOmega(n)$ and $\omega(n)$ denote the number of distinct prime factors of the positive integer $n$, counted respectively with and without multiplicity. Let $d_k(n)$ 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 \[ \sum_{n\leq x,\varOmega(n)-\omega(n)=q}f(n) \] for a class of multiplicative functions $f$, including in particular $f(n)=d_k(n)$, unconditionally if $1\leq k\leq 3$, and under some reasonable assumptions if $k\geq 4$.
The result also applies to $f(n)={\varphi(n)}/{n}$ (where $\varphi$ is the totient function), to $f(n)={\sigma_r(n)}/{n^r}$ (where $\sigma_r$ is the sum of $r$th powers of divisors) and to functions related to the notion of exponential divisor. It generalizes similar results by J. Wu and Y.-K. Lau when $f(n)=1$, respectively $f(n)=d_2(n)$.
