Additive functionals of Harmonic samples: the conditioned Dickman regime

TL;DR

This paper investigates the distribution of additive arithmetic functions evaluated at harmonic samples, revealing convergence to conditioned Dickman limits, contrasting with classical Gaussian results for uniform samples, using probabilistic and analytic methods.

Contribution

It introduces a new understanding of additive functions on harmonic samples, showing convergence to Dickman-type limits, and combines probabilistic and analytic techniques for this analysis.

Findings

Additive functions on harmonic samples converge to conditioned Dickman distributions.

Contrasts with Gaussian limits in classical Erdős-Kac theorem.

Uses probabilistic representation and analytic tools like Mertens' approximation.

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 Erd\"os-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