2024/06/13 by Tony Haddad, Dimitris Koukoulopoulos, Haddad, Tony +1 · 1 citation
Computer Science · Mathematics · #11N25 #11N37 #11N60 #60B12 #Bayesian Methods and Mixture Models #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2406.09360
openalex publication_date 2024/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let x ≥ 2, let Nx be an integer chosen uniformly at random from the set \mathbb Z ∩ [1, x], and let (V1, V2, …) be a Poisson--Dirichlet process of parameter 1. We prove that there exists a coupling of these two random objects such that \mathbb E ∑i ≥ 1 |log Pi- Vilog x| \asymp 1, where the implied constants are absolute and Nx = P1P2 ⋯ is the unique factorization of Nx into primes or ones with the Pi's being non-increasing. This establishes a 2002 conjecture of Arratia arXiv:1305.0941 who constructed a coupling for which the left-hand side in the above estimate is ≪ log log x, and who also proved that the left-hand side is ≥ 1-o(1) for all couplings. In addition, we use our refined coupling to give a probabilistic proof of the Dirichlet law for the average distribution of the integer factorization into k parts proved in 2023 by Leung arXiv:2206.14728 and we improve on its error term.