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

TL;DR

This paper introduces a new inequality and strategy to asymptotically count integers with divisors in a given interval, extending previous probabilistic methods in number theory.

Contribution

It develops a general approach combining inequalities and couplings to derive asymptotic formulas for divisor counting problems in number theory.

Findings

Derived an asymptotic formula for integers with divisors in (y, z) as z/y → ∞

Established a probabilistic framework reducing divisor counting to boundary probability bounds

Applied the method to large interval regimes in number theory

Abstract

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 $\ell^1(\mathbb R)$, 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 $z/y \to \infty$ as $x \to \infty$.