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Sur les processus arithmétiques liés aux diviseurs

Part of:Probabilistic theory: distribution modulo $1$; metric theory of algorithmsMultiplicative number theory

Published online by Cambridge University Press:  25 July 2016

R. de la Bretèche and
G. Tenenbaum
R. de la Bretèche*
Affiliation:
Université Paris Diderot‒Paris 7
G. Tenenbaum*
Affiliation:
Université de Lorraine
*
Université Paris Diderot--Paris 7, Sorbonne Paris Cité, UMR 7586, Institut de Mathématiques de Jussieu--PRG, Case 7012, F-75013 Paris, France. Email address: regis.delabreteche@imj-prg.fr
Institut Élie Cartan, Université de Lorraine, BP 70239, F-54506 Vandoeuvre-lès-Nancy Cedex, France. Email address: gerald.tenenbaum@univ-lorraine.fr
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Abstract

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For natural integer n, let D n denote the random variable taking the values log d for d dividing n with uniform probability 1/τ(n). Then t↦ℙ(D n ≤n t ) (0≤t≤1) is an arithmetic process with respect to the uniform probability over the first N integers. It is known from previous works that this process converges to a limit law and that the same holds for various extensions. We investigate the generalized moments of arbitrary orders for the limit laws. We also evaluate the mean value of the two-dimensional distribution function ℙ(D n ≤n u , D{n/D n}≤n v ).

Keywords

Arc-sine lawarithmetic modelarithmetic processDirichlet lawsdivisors

MSC classification

Primary:11K65: Arithmetic functions
Secondary:11N37: Asymptotic results on arithmetic functions11N64: Other results on the distribution of values or the characterization of arithmetic functions

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue A: Probability, Analysis and Number Theory , July 2016, pp. 63 - 76
Copyright
Copyright © Applied Probability Trust 2016 

References

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