A subset generalization of the Erd\H{o}s-Kac theorem over number fields with applications

TL;DR

This paper extends the Erdős-Kac theorem to subsets of ideals in number fields, providing new probabilistic results and applications to specific classes of ideals such as $h$-free and $h$-full ideals.

Contribution

It generalizes the Erdős-Kac theorem to subsets of ideals in number fields under certain conditions, broadening its scope and applications.

Findings

Established Gaussian distribution for subsets of ideals in number fields.

Proved Erdős-Kac theorem for $h$-free and $h$-full ideals.

Extended previous generalizations to new ideal subsets.

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. In 2004, the third author generalized their theorem to all abelian monoids. In this work, we extend the work of the third author to any subset of the set of ideals of a number field satisfying some additional conditions. Finally, we apply this theorem to prove the Erd\H{o}s-Kac theorem over $h$-free and over $h$-full ideals of the number field.