Distribution of ω(n) over h-free and h-full numbers

Let [Formula: see text] denote the number of distinct prime factors of a natural number [Formula: see text]. In 1917, Hardy and Ramanujan proved that [Formula: see text] has normal order [Formula: see text] over naturals. In this work, we establish the first and second moments of [Formula: see text] over [Formula: see text]-free and [Formula: see text]-full numbers using a new counting argument and prove that [Formula: see text] has normal order [Formula: see text] over these subsets.