Year: 2026
Type: article
Abstract: Let $\omega (n)$ denote the number of distinct prime factors of a natural number n . In 1940, Erdős and Kac established that $\omega (n)$ obeys the Gaussian distribution over natural numbers, and in 2004, the third author generalized their theorem to all abelian monoids. In this article, 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.
Source: Canadian Journal of Mathematics
Authors: Sourabhashis Das, Wentang Kuo, Yu-Ru Liu
Institution: University of Waterloo
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W7128444601