Additive functionals of Harmonic samples: the conditioned Dickman regime

2025/12/02 by Victor Bernal Ramirez, Ramirez, Victor Bernal, Arturo Jaramillo +1
Computer Science · Economics, Econometrics and Finance · Physics and Astronomy · #Bayesian Methods and Mixture Models #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR) #Statistical Mechanics and Entropy #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2512.02926

openalex publication_date 2025/12/02 · openalex created_date 2025/12/04 · openalex updated_date 2026/07/28

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.

Citations

• Approximation of Smooth Numbers for Harmonic Samples A Stein method Approach
• Dickman Approximation of weighted random sums in the Kolmogorov distance
• A probabilistic approach to the Erdös-Kac theorem for additive functions

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