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arXiv · 2512.02926

Additive functionals of Harmonic samples: the conditioned Dickman regime

Victor Bernal Ramirez·Arturo Jaramillo
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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 Erd\"os-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.

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Victor Bernal Ramirez, Arturo Jaramillo. 2025-12-02. Additive functionals of Harmonic samples: the conditioned Dickman regime. https://arxiv.org/abs/2512.02926

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