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

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Zurita R.

Acta Arithmetica

2013

tom 161

nr 1

69-100

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

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1 Acta Arithmetica

1 Zurita R.

1 2013

Wyniki wyszukiwania

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