Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T06:09:22.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Sur les processus arithmétiques liés aux diviseurs

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Multiplicative 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
Get access Rights & Permissions [Opens in a new window]

Abstract

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 functions 11N64: Other results on the distribution of values or the characterization of arithmetic functions
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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

[1] Babu, G. J. (1973).A note on the invariance principle for additive functions.Sankhyā A 35,307–310.Google Scholar
[2] Bareikis, G. et Mačiulis, A. (2012).Cesàro means related to the square of the divisor function.Acta Arithmetica 156,83–99.CrossRefGoogle Scholar
[3] Bareikis, G. et Mačiulis, A. (2015).On the second moment of an arithmetical process related to the natural divisors.Ramanujan J. 37,1–24.CrossRefGoogle Scholar
[4] Bareikis, G. et Manstavičius, E. (2007).On the DDT theorem.Acta Arithmetica 126,155–168.CrossRefGoogle Scholar
[5] Billingsley, P. (1970).Additive functions and Brownian motion.Notices Amer. Math. Soc. 17, 1050.Google Scholar
[6] Billingsley, P. (1974).The probability theory of additive arithmetic functions.Ann. Prob. 2,749–791.CrossRefGoogle Scholar
[7] Deshouillers, J.-M.,Dress, F. et Tenenbaum, G. (1979).Lois derépartition des diviseurs, I.Acta Arithmetica 34,273–285.CrossRefGoogle Scholar
[8] Erdős, P. et Kac, M. (1947).On the number of positive sums of independent random variables.Bull. Amer. Math. Soc. 53,1011–1020.CrossRefGoogle Scholar
[9] Feller, W. (1968).An Introduction to Probability Theory and Its Applications, Vol. 1,3rd edn.John Wiley,New York.Google Scholar
[10] Ferguson, T. S. (1973).A Bayesian analysis of some nonparametric problems.Ann. Statist. 1,209–230.CrossRefGoogle Scholar
[11] Hall, R. R. (1996).Sets of Multiples (Camb. Tracts Math. 118).Cambridge University Press.CrossRefGoogle Scholar
[12] Hall, R. R. et Tenenbaum, G. (1988).Divisors (Camb. Tracts Math. 90).Cambridge University Press.CrossRefGoogle Scholar
[13] Hirth, U. M. (1997).Probabilistic number theory, the GEM/Poisson-Dirichlet distribution and the arc-sine law.Combinatorics Prob. Comput. 6,57–77.CrossRefGoogle Scholar
[14] Kubilius, J. (1956).Méthodes probabilistes en théorie des nombres.Uspekhi Mat. Nauk 11,31–66 (en russe). Traduction anglaise: Amer. Math. Soc. Transl. Ser. 2 19 (1962),47–85.Google Scholar
[15] Kubilius, J. (1964).Probabilistic Methods in the Theory of Numbers (Transl. Math. Monogr. 11).American Mathematical Society,Providence, RI.CrossRefGoogle Scholar
[16] Manstavičius, E. (1988).An invariance principle for additive arithmetic functions.Dokl. Akad. Nauk 298,1316–1320 (en russe). Traduction anglaise: Soviet Math. Dokl. 37 (1988),259–263.Google Scholar
[17] Manstavičius, E. (1996).Natural divisors and the Brownian motion.J. Théor. Nombres Bordeaux 8,159–171.CrossRefGoogle Scholar
[18] Manstavičius, E. et Timofeev, N. M. (1997).A functional limit theorem related to natural divisors.Acta Math. Hung. 75,1–13.CrossRefGoogle Scholar
[19] Nyandwi, S. et Smati, A. (2013).Distribution laws of pairs of divisors.Integers 13, A13.Google Scholar
[20] Philipp, W. (1973).Arithmetic functions and Brownian motion. Dans Analytic Number Theory (St. Louis 1972) (Proc. Sympos. Pure Math. 24),American Mathematical Society,Providence, RI, pp. 233–246.CrossRefGoogle Scholar
[21] Tenenbaum, G. (1979).Lois de répartition des diviseurs, 4.Ann. Inst. Fourier (Grenoble) 29,1–15.CrossRefGoogle Scholar
[22] Tenenbaum, G. (1997).Addendum to the paper of E. Manstavičius & M. N. Timofeev ``A functional limit theorem related to natural divisors''.Acta Math. Hung. 75,15–22.CrossRefGoogle Scholar
[23] Tenenbaum, G. (2015).Introduction à la théorie analytique et probabiliste des nombres, 4e édition.Collection Échelles,Belin, Paris. Traduction anglaise: Introduction to Analytic and Probabilistic Number Theory (Graduate Stud. Math. 163),3rd edn.,American Mathematical Society,Providence, RI, 2015.Google Scholar
[24] Timofeev, N. M. et Usmanov, Kh. Kh. (1982).Arithmetic modelling of Brownian motion.Dokl. Akad. Nauk Tadzhik. SSR 25,207–211 (en russe).Google Scholar