Generalizations of Erdős-Kac theorem with applications

2025/06/03 by Sourabhashis Das, Wentang Kuo, Das, Sourabhashis +3 · 1 citation
Earth and Planetary Sciences · Mathematics · #11K65 #11N80 #20M32 #Advanced Algebra and Geometry #FOS: Mathematics #Mathematical Analysis and Transform Methods #Number Theory (math.NT) #Probability (math.PR) #Seismic Imaging and Inversion Techniques

paper · pdf · doi:10.48550/arxiv.2506.02432

openalex publication_date 2025/06/03 · openalex created_date 2025/10/07 · openalex updated_date 2026/07/30

Abstract

Let ω(n) denote the number of distinct prime factors of a natural number n. In 1940, Erdős and Kac established that ω(n) obeys the Gaussian distribution over natural numbers, and in 2004, the third author generalized their theorem to all abelian monoids. In this paper, we extend her theorem to any subsets of an abelian monoid satisfying some additional conditions, and apply this result to the subsets of h-free and h-full elements. We study generalizations of several arithmetic functions, such as the prime counting omega functions and the divisor function in a unified framework. Finally, we apply our results to number fields, global function fields, and geometrically irreducible projective varieties, demonstrating the broad relevance of our approach.

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• On the distribution of the total number of generators of h-free and h-full elements in an abelian monoid

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