Sur les processus arithmétiques liés aux diviseurs. (July 2016)

• Record Type: Journal Article
• Authors: de la Bretèche, R.; Tenenbaum, G.
• Editors: Goldie, C. M.; Mijatović, A.
• Is Part Of: Advances in applied probability. Volume 48:Number A(2016)
• Page Start: 63
• Page End: 76
• Publication Date: 2016-07
• Subjects: Arc-sine law, -- arithmetic model, -- arithmetic process, -- Dirichlet laws, -- divisors; Primary 11K65, -- Secondary 11N37, -- 11N64; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2016.42
• Languages: English
• ISSNs: 0001-8678

Abstract

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 ).