Attenuated Poisson Dirichlet approximations for divisibility configurations

We study the point process formed by the normalized logarithms of the distinct prime factors of a harmonic random sample.

We prove a quantitative convergence result, in a Wasserstein-type metric over decreasing sequences, toward the atom sequence of a Dickman Poisson cloud conditioned to have total mass at most one, equivalently a uniformly attenuated Poisson-Dirichlet law.

The proof is based on the conditioned geometric representation of harmonic samples, a Poisson approximation chain for the associated point processes, monotone couplings of Poisson point processes, and Kolmogorov estimates for the Dickman approximation of weighted geometric sums.