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# Sur les processus arithmétiques liés aux diviseurs

## 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 ).

## Corresponding author

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

## References

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[1] Babu, G. J. (1973).A note on the invariance principle for additive functions.Sankhyā A 35,307310.
[2] Bareikis, G. et Mačiulis, A. (2012).Cesàro means related to the square of the divisor function.Acta Arithmetica 156,8399.
[3] Bareikis, G. et Mačiulis, A. (2015).On the second moment of an arithmetical process related to the natural divisors.Ramanujan J. 37,124.
[4] Bareikis, G. et Manstavičius, E. (2007).On the DDT theorem.Acta Arithmetica 126,155168.
[5] Billingsley, P. (1970).Additive functions and Brownian motion.Notices Amer. Math. Soc. 17, 1050.
[6] Billingsley, P. (1974).The probability theory of additive arithmetic functions.Ann. Prob. 2,749791.
[7] Deshouillers, J.-M.,Dress, F. et Tenenbaum, G. (1979).Lois derépartition des diviseurs, I.Acta Arithmetica 34,273285.
[8] Erdős, P. et Kac, M. (1947).On the number of positive sums of independent random variables.Bull. Amer. Math. Soc. 53,10111020.
[9] Feller, W. (1968).An Introduction to Probability Theory and Its Applications, Vol. 1,3rd edn.John Wiley,New York.
[10] Ferguson, T. S. (1973).A Bayesian analysis of some nonparametric problems.Ann. Statist. 1,209230.
[11] Hall, R. R. (1996).Sets of Multiples (Camb. Tracts Math. 118).Cambridge University Press.
[12] Hall, R. R. et Tenenbaum, G. (1988).Divisors (Camb. Tracts Math. 90).Cambridge University Press.
[13] Hirth, U. M. (1997).Probabilistic number theory, the GEM/Poisson-Dirichlet distribution and the arc-sine law.Combinatorics Prob. Comput. 6,5777.
[14] Kubilius, J. (1956).Méthodes probabilistes en théorie des nombres.Uspekhi Mat. Nauk 11,3166 (en russe). Traduction anglaise: Amer. Math. Soc. Transl. Ser. 2 19 (1962),4785.
[15] Kubilius, J. (1964).Probabilistic Methods in the Theory of Numbers (Transl. Math. Monogr. 11).American Mathematical Society,Providence, RI.
[16] Manstavičius, E. (1988).An invariance principle for additive arithmetic functions.Dokl. Akad. Nauk 298,13161320 (en russe). Traduction anglaise: Soviet Math. Dokl. 37 (1988),259263.
[17] Manstavičius, E. (1996).Natural divisors and the Brownian motion.J. Théor. Nombres Bordeaux 8,159171.
[18] Manstavičius, E. et Timofeev, N. M. (1997).A functional limit theorem related to natural divisors.Acta Math. Hung. 75,113.
[19] Nyandwi, S. et Smati, A. (2013).Distribution laws of pairs of divisors.Integers 13, A13.
[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. 233246.
[21] Tenenbaum, G. (1979).Lois de répartition des diviseurs, 4.Ann. Inst. Fourier (Grenoble) 29,115.
[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. 75,1522.
[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.
[24] Timofeev, N. M. et Usmanov, Kh. Kh. (1982).Arithmetic modelling of Brownian motion.Dokl. Akad. Nauk Tadzhik. SSR 25,207211 (en russe).

# Sur les processus arithmétiques liés aux diviseurs

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