Sur les processus arithmétiques liés aux diviseurs

Abstract

AbstractFor natural integer n, let Dn denote the random variable taking the values log d for d dividing n with uniform probability 1/τ(n). Then t↦ℙ(Dn≤nt) (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(Dn≤nu, D{n/Dn}≤nv).