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The Ramanujan Journal

, Volume 9, Issue 1–2, pp 139–202 | Cite as

  • Régis De La Bretèche
  • Gérald Tenenbaum
Article

Abstract

Let Ψ(x,y) (resp. Ψ 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 Ψ m (x/d,y)/Ψ(x,y) for 1≤slantdslantx, m≥slant 1, x≥slant y≥slant 2, turns out to be a crucial step for estimating arithmetic 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.

Keywords

friable numbers saddle point method arithmetic functions 

Propriétés statistiques des entiers friables

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Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.École Normale Supérieure, Département de Mathématiques et ApplicationsParis Cedex 05France
  2. 2.Institut Élie CartanUniversité de Nancy 1Vandœuvre CedexFrance

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