Generalizations of Erd\H{o}s-Kac theorem with applications

TL;DR

This paper extends the Erd ext{"o}s-Kac theorem to broader subsets of abelian monoids, applies it to various arithmetic functions, and demonstrates its relevance across number fields, function fields, and algebraic varieties.

Contribution

It generalizes the Erd ext{"o}s-Kac theorem to new subsets of abelian monoids and applies this to diverse mathematical contexts.

Findings

Gaussian distribution of prime factors in new subsets

Unified framework for arithmetic functions

Applications to number fields and algebraic varieties

Abstract

Let $\omega(n)$ denote the number of distinct prime factors of a natural number $n$. In 1940, Erd\H{o}s 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 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.