On Arratia's coupling and the Dirichlet law for the factors of a random integer

Let x ≥ 2, let Nₓ be an integer chosen uniformly at random from the set 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 E \, ∑i ≥ 1 |log Pᵢ- Vᵢlog x| 1, where the implied constant is absolute and Nₓ = P₁P₂ ⋯ is the unique factorization of Nₓ into primes or ones with the Pᵢ's being non-increasing. This establishes a conjecture of Arratia (2002) arXiv:1305.0941 who proved that the right-hand side in the above estimate can be made log\!log x. We also use this 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.