Distribution of $\omega(n)$ over $h$-free and $h$-full numbers

TL;DR

This paper investigates the distribution of the number of distinct prime factors, (n), over special subsets of natural numbers called h-free and h-full numbers, establishing their moments and confirming the normal order (n) follows   n.

Contribution

It introduces a new counting method to analyze (n) over h-free and h-full numbers and proves the normal order result for these subsets.

Findings

First and second moments of (n) over h-free and h-full numbers are established.

(n) has normal order    n over these subsets.

The work extends classical results to specialized number subsets.

Abstract

Let $\omega(n)$ denote the number of distinct prime factors of a natural number $n$. In 1917, Hardy and Ramanujan proved that $\omega(n)$ has normal order $\log \log n$ over naturals. In this work, we establish the first and the second moments of $\omega(n)$ over $h$-free and $h$-full numbers using a new counting argument and prove that $\omega(n)$ has normal order $\log \log n$ over these subsets.