Extensions of Billingsley's Theorem via Multi-Intensities

Abstract

Let $p_1 \ge p_2 \ge \dots$ be the prime factors of a random integer chosen uniformly from $1$ to $n$, and let $$ \frac{\log p_1}{\log n}, \frac{\log p_2}{\log n}, \dots $$ be the sequence of scaled log factors. Billingsley's Theorem (1972), in its modern formulation, asserts that the limiting process, as $n \to \infty$, is the Poisson-Dirichlet process with parameter $\theta =1$. In this paper we give a new proof, inspired by the 1993 proof by Donnelly and Grimmett, and extend the result to factorizations of elements of normed arithmetic semigroups satisfying certain growth conditions, for which the limiting Poisson-Dirichlet process need not have $\theta =1$. We also establish Poisson-Dirichlet limits, with $\theta \ne 1$, for ordinary integers conditional on the number of prime factors deviating from the usual value $\log \log n$. At the core of our argument is a purely probabilistic lemma giving a new criterion for convergence in distribution to a Poisson-Dirichlet process, from which the number-theoretic applications follow as straightforward corollaries. The lemma uses ingredients similar to those employed by Donnelly and Grimmett, but reorganized so as to allow subsequent number theory input to be processed as rapidly as possible. A by-product of this work is a new characterization of Poisson-Dirichlet processes in terms of multi-intensities.