Natural divisors and the brownian motion

Manstavičius, Eugenijus

Journal de théorie des nombres de Bordeaux, Tome 8 (1996) no. 1, pp. 159-171.

On propose un modèle du mouvement brownien relatif aux diviseurs d’un entier, et on établit la convergence faible de la mesure associée dans l’espace $𝐃[0,1]$. On obtient un résultat analogue à celui obtenu par Erdös pour les diviseurs premiers [6] (cf. [14] pour une démonstration). Ces résultats et les recherches de l’auteur [15] étendent l’étude [9] de la distribution des diviseurs. Notre approche s’appuie sur les théorèmes limites fonctionnels en théorie des probabilités.

A model of the Brownian motion defined in terms of the natural divisors is proposed and weak convergence of the related measures in the space $𝐃$ [0,1] is proved. An analogon of the Erdös arcsine law, known for the prime divisors [6] (see [14] for the proof), is obtained. These results together with the author’s investigation [15] extend the systematic study [9] of the distribution of natural divisors. Our approach is based upon the functional limit theorems of probability theory.

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