On the number of prime factors with a given multiplicity over h-free and h-full numbers

TL;DR

This paper investigates the distribution of prime factors with specific multiplicities over h-free and h-full numbers, establishing normal orders and probabilistic behaviors for these functions.

Contribution

It provides asymptotic estimates and proves normal order results for the number of prime factors with fixed multiplicity over specialized number sets, extending classical results.

Findings

(n) has normal order  erences over h-free numbers

(n) has normal order  erences over h-full numbers

(n) satisfies the Erds-Kac Theorem

Abstract

Let $k$ and $n$ be natural numbers. Let $\omega_k(n)$ denote the number of distinct prime factors of $n$ with multiplicity $k$ as studied by Elma and the third author. We obtain asymptotic estimates for the first and the second moments of $\omega_k(n)$ when restricted to the set of $h$-free and $h$-full numbers. We prove that $\omega_1(n)$ has normal order $\log \log n$ over $h$-free numbers, $\omega_h(n)$ has normal order $\log \log n$ over $h$-full numbers, and both of them satisfy the Erd\H{o}s-Kac Theorem. Finally, we prove that the functions $\omega_k(n)$ with $1 < k < h$ do not have normal order over $h$-free numbers and $\omega_k(n)$ with $k > h$ do not have normal order over $h$-full numbers.