Sourabhashis Das, University of Waterloo
On the distributions of prime divisor counting function
In 1917, Hardy and Ramanujan established that $\omega(n)$, the number of distinct prime factors of a natural number $n$, and $\Omega(n)$, the total number of prime factors of $n$ have normal order $\log \log n$. In 1940, Erdős and Kac refined this understanding by proving that $\omega(n)$ follows a Gaussian distribution over the natural numbers.
In this talk, we extend these classical results to the subsets of $h$-free and $h$-full numbers. We show that $\omega_1(n)$, the number of distinct prime factors of $n$ with multiplicity exactly $1$, has normal order $\log \log n$ over $h$-free numbers. Similarly, $\omega_h(n)$, the number of distinct prime factors with multiplicity exactly $h$, has normal order $\log \log n$ over $h$-full numbers. However, for $1 < k < h$, we prove that $\omega_k(n)$ does not have a normal order over $h$-free numbers, and for $k > h$, $\omega_k(n)$ does not have a normal order over $h$-full numbers.
Furthermore, we establish that $\omega_1(n)$ satisfies the Erdős-Kac theorem over $h$-free numbers, while $\omega_h(n)$ does so over $h$-full numbers. These results provide a deeper insight into the distribution of prime factors within structured subsets of natural numbers, revealing intriguing asymptotic behavior in these settings.
MC 5417