Additive functionals of Harmonic samples: the conditioned Dickman regime

Abstract

We study the distributional behavior of additive arithmetic functions evaluated at integers drawn from the harmonic distribution. Our main result shows that, for a broad family of completely additive functions, their evaluations at harmonic samples, suitably normalized, converge in law to conditioned Dickman-type Poisson integrals. This behavior contrasts with the Gaussian limits arising in the classical Erdös-Kac theorem under uniform sampling. Our approach combines the probabilistic representation of harmonic samples via independent geometric variables, analytic inputs such as Mertens' approximation, and a Poissonization procedure.