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

Rimer Zurita

Acta Arithmetica, Tome 161 (2013), p. 69-100 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $d_k(n)$ 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 $∑_{n≤x,Ω(n)-ω(n)=q} f(n)$ for a class of multiplicative functions f, including in particular $f(n) = d_k(n)$, 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 function), to $f(n) = σ_r(n)/(n^r)$ (where $σ_r$ is the sum of rth 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

Publié le : 2013-01-01

Zbl 1278.11092

EUDML-ID : urn:eudml:doc:278958