Additive functionals of Harmonic samples: the conditioned Dickman regime
Victor Bernal Ramirez, Arturo Jaramillo
TL;DR
The paper analyzes additive arithmetic functions evaluated at harmonic-random integers and shows that, for a broad class of φ, φ(H_n) converges in law to a Dickman-type distribution conditioned on small values. The authors leverage a probabilistic representation of harmonic samples via independent geometric variables, a Poissonization framework, and Mertens-type prime-sum estimates to derive the limit as a Poisson-integral世界-based object (a Dickman subordinator) conditioned on a small total. The key result is a new conditioned Dickman limit in the harmonic setting, contrasting with the Gaussian Erdős–Kac behavior under uniform sampling, and it relies on a precise decomposition into Poisson integrals and careful analytic control. This advances probabilistic number theory by extending Dickman-type phenomena to harmonic sampling and provides a framework for identifying and proving conditioned infinite-divisibility limits for additive functions.
Abstract
We study the distributional behavior of additive arithmetic functions evaluated at integers drawn from the harmonic distribution. Our main result shows that a broad family of such functions converges in law to conditioned Dickman-type limits. This contrasts with the Gaussian limits of the classical Erdös-Kac theorem for uniform samples. Our perspective combines the probabilistic representation of harmonic samples via independent geometric variables with analytic inputs such as Mertens' approximation, together with a Poissonization procedure