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Sur la probabilité qu’un entier possède un diviseur dans un intervalle donné. (French) Zbl 0541.10038
The number H(x,y,z) of positive integers \(n<x\) with at least one divisor d satisfying \(y\leq d<z\) for certain y,z dependent on x has been studied in earlier papers, including, for example, [Acta Arith. 38, 1-36 (1980; Zbl 0437.10028)] by the present author. He continues this investigation here by establishing upper and lower bounds for \(x^{-1} H(x,y,z)\) in terms of functions of u, where \(z=y^{1+u}\), in the case when \(1<2y\leq z\leq \min(y^{3/2},x^{1/2}).\) He also considers the cases \(z<2y\), log y\(=o(\log z)\).
Reviewer: E.J.Scourfield

MSC:
11N37 Asymptotic results on arithmetic functions
11B83 Special sequences and polynomials
11K65 Arithmetic functions in probabilistic number theory
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References:
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