Propriétés statistiques des entiers friables

Régis de La Bretèche (1) , Gérald Tenenbaum (2)

1 IMJ - Institut de Mathématiques de Jussieu
2 IECN - Institut Élie Cartan de Nancy

Let Psi(x,y) (resp. Psi_m(x,y)) denote the number of integers not exceeding x that are y-friable, i.e., have no prime factor exceeding y (resp. and are coprime to m). Evaluating the ratio Psi_m(x/d,y)/Psi(x,y) for 1<=d<=x, m>=1, x<=y<=2, turns out to be a crucial step for estimating arithmetical sums over friable integers. Here, it is crucial to obtain formulae with a very wide range of validity. In this paper, several uniform estimates are provided for the aforementioned ratio, which supersede all previously known results. Applications are given to averages of various arithmetic functions over friable integers which in turn improve corresponding results from the literature. The technique employed rests mainly on the saddle-point method, which is an efficient and specific tool for the required design.

Mots clés: sieve; friable integers; integers free of large prime facctors; saddle point method; mean values of arithmetic functions

Domaines: Théorie des nombres [math.NT]

Régis de La Bretèche, Gérald Tenenbaum. Propriétés statistiques des entiers friables. Ramanujan Journal (The), 2005, 9, pp.139-202. ⟨hal-00079665⟩