Poisson-Dirichlet approximation for counting integers with divisors in an interval

We give a simple inequality that compares the laws of two random variables taking values in a convex subset of a normed vector space. By combining this with Arratia’s coupling, recently refined by Koukoulopoulos and the author, we obtain a general strategy to reduce the problem of finding an asymptotic formula for the number of integers whose prime factorization lies in any given subset of ${\mathrm{\ell }}^1(\mathbb{R})$, to bounding two key probabilities measuring proximity to the boundary of the subset in question. We apply this strategy to obtain an asymptotic formula for counting integers in $[1,x]$ that have a divisor in an interval $(y,z)$ in the regime $z\mathrm{/}y\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$.