Distribution Laws of Pairs of Divisors

In this paper we study the distribution of pairs (d1, d2) of positive integers such that the product d1d2 divides a given integer n from a probabilistic point of view. The number of these pairs, denoted by τ3(n), is equal to the number of ways to write n as a product of three positive integers. To these pairs we associate a random vector taking the values ( (log d1)/(log n), (log d2)/(log n) ) with uniform probability 1/τ3(n) and its distribution function Fn. We show that the mean of Fn uniformly converges to the distribution function of the Beta two-dimensional law ( Dirichlet law). Our study generalizes a work done by Deshouillers, Dress and Tenenbaum in the case of the divisors of an integer where they showed that the average distribution of divisors of a given integer follows the arcsine law.