The sizes of large prime factors for a random integer N sam- pled uniformly in [1, x] are known to converge in distribution to a Poisson- Dirichlet process V = (V1, V2, . . .) as x → ∞. In 2002, Arratia constructed a coupling of N and V satisfying E 􏰇i | log Pi − (log x)Vi| = O(log log x) where P1P2 ··· is the unique factorization of N with P1 ≥ P2 ≥ ··· being all primes or ones. He conjectured that there exists a coupling for which this expectation is O(1).

I will present a modification of his coupling which proves his conjecture, and show that O(1) is optimal. As a corollary, we deduce the arcsine law in the average distribution of divisors proved by Deshouillers, Dress and Tenenbaum in 1979. This is joint work with Dimitris Koukoulopoulos.