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

• Autorzy: Rimer Zurita
• Języki publikacji: FR EN

Abstrakty

EN

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

Kategorie tematyczne

• 11N37: Asymptotic results on arithmetic functions
• 11N25: Distribution of integers with specified multiplicative constraints
• 11M26: Nonreal zeros of ζ ( s ) and L ( s , χ ) ; Riemann and other hypotheses
• 11N60: Distribution functions associated with additive and positive multiplicative functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 161
• Numer: 1
• Strony: 69-100

Daty

wydano

2013

Twórcy

autor

Rimer Zurita

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

Identyfikatory

DOI

10.4064/aa161-1-5