Propriétés statistiques des entiers friables |
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Authors: | Régis De La Bretèche Gérald Tenenbaum |
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Institution: | (1) École Normale Supérieure, Département de Mathématiques et Applications, 45, rue d’Ulm, 75230 Paris Cedex 05, France;(2) Institut Élie Cartan, Université de Nancy 1, BP 239, 54506 Vandœuvre Cedex, France |
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Abstract: | Let Ψ(x,y) (resp. Ψm(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 Ψm(x/d,y)/Ψ(x,y) for 1≤slantd≤slantx, 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.2000 Mathematics Subject Classification: Primary—11N25; Secondary—11K65, 11N37 |
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Keywords: | friable numbers saddle point method arithmetic functions |
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