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

Abstract

Let $x \ge 2$, let $N_x$ be an integer chosen uniformly at random from the set $\mathbb Z \cap [1, x]$, and let $(V_1, V_2, \ldots)$ be a Poisson--Dirichlet process of parameter $1$. We prove that there exists a coupling of these two random objects such that $$ \mathbb E \, \sum_{i \ge 1} |\log P_i- V_i\log x| \asymp 1, $$ where the implied constants are absolute and $N_x = P_1P_2 \cdots$ is the unique factorization of $N_x$ into primes or ones with the $P_i$'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 $\ll \log\!\log x$, and who also proved that the left-hand side is $\ge 1-o(1)$ 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.