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Moyennes de certaines fonctions multiplicatives sur les entiers friables
- Source :
- Journal für die reine und angewandte Mathematik, Journal für die reine und angewandte Mathematik, 2003, 564, pp.119-166, Journal für die reine und angewandte Mathematik, Walter de Gruyter, 2003, 564, pp.119-166
- Publication Year :
- 2003
- Publisher :
- HAL CCSD, 2003.
-
Abstract
- International audience; An integer n is called friable when its prime number factorization consists exclusively of (relatively) small factors. Let P(n) be the largest prime factor occurring in this factorization, and let S(x,y) denote the set {n? x,P(n)? y}. This work investigates the asymptotic behaviour of the summatory function Psi_f(x, y):=\sum_{n\in S(x,y)}f(n) when f is a multiplicative arithmetical function satisfying some simple and general conditions concerning its mean behaviour on primes and on powers of primes. One such condition is |\sum_{p? z}f(p)\log p-\kappa z|? Cz/R(z) (z>1), where C is a constant and \kappa>0; requirements on R are technical conditions too long to state here, but satisfied by any "reasonable" positive increasing function. By setting R(z)=(\log z)^\delta in the very general Théorème 2.1, the authors obtain a more general as well as more precise estimate on \Psi_f(x, y) (Corollaire 2.2) than that recently obtained by J. M. Song [Acta Arith. 102 (2002), no. 2, 105--129; MR1889623 (2003a:11123)]. Their next result (Corollaire 2.3) offers a general estimate in the case where f(p) is on average very close to a constant, and contains without loss of precision estimates of the literature for particular functions f, such as the so-called Piltz functions \tau_k, or the function µ^2 where µ is the Möbius function. Then, as a further application of their first result, they establish an Erdös-Wintner theorem on friable integers (Théorème 2.4). They finally mention an application to the case where f(n) is the characteristic function of the integers that can be represented as a sum of two squares of integers (Théorème 2.5); their estimate is uniformly valid for x ? 3, exp((\log x)^{2/5+\epsilon})? y? x. The paper begins with an historical introduction, and is followed by an extensive bibliography on the subject.
- Subjects :
- friable integers
Applied Mathematics
General Mathematics
010102 general mathematics
Multiplicative function
Prime number
Fermat's theorem on sums of two squares
multiplicative function
010103 numerical & computational mathematics
Function (mathematics)
mean values of arithmetic functions
Möbius function
01 natural sciences
[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]
Combinatorics
Factorization
Prime factor
AMS Classification: 11N37, 11N25
Calculus
Arithmetic function
0101 mathematics
[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]
Mathematics
Erdös-Wintner theorem
Subjects
Details
- Language :
- French
- ISSN :
- 00754102 and 14355345
- Database :
- OpenAIRE
- Journal :
- Journal für die reine und angewandte Mathematik, Journal für die reine und angewandte Mathematik, 2003, 564, pp.119-166, Journal für die reine und angewandte Mathematik, Walter de Gruyter, 2003, 564, pp.119-166
- Accession number :
- edsair.doi.dedup.....a91a2f9cd395c675c3c3f23001187ba8