Sur un problème de Rényi et Ivić concernant les fonctions de diviseurs de Piltz

Rimer Zurita¹

¹ Section de Mathématiques Université de Genève Case postale 64 2-4, rue du Lièvre 1211 Genève 4, Suisse

Acta Arithmetica, Tome 161 (2013) no. 1, pp. 69-100

Résumé

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 $f(n)=d_2(n)$.

Détail

DOI : 10.4064/aa161-1-5

Mots-clés : varomega omega denote number distinct prime factors positive integer counted respectively without multiplicity denote piltz function which counts number ways writing product factors obtain precise estimate sum sum leq varomega omega class multiplicative functions including particular unconditionally leq leq under reasonable assumptions geq result applies varphi where varphi totient function sigma where sigma sum rth powers divisors functions related notion exponential divisor generalizes similar results k lau respectively