On the distributions of prime divisor counting functions

Abstract

Let k and n be natural numbers. Let ω(n) denote the number of distinct prime factors of n, Ω(n) denote the total number of prime factors of n counted with multiplicity, and ω_k(n) denote the number of distinct prime factors of n that occur with multiplicity exactly k.

Let h ≥ 2 be a natural number. We say that n is h-free if every prime factor of n has multiplicity less than h, and h-full if all prime factors of n have multiplicity at least h.

In 1917, Hardy and Ramanujan proved that both ω(n) and Ω(n) have normal order log log n over the natural numbers. In this thesis, using a new counting argument, we establish the first and second moments of all these arithmetic functions over the sets of h-free and h-full numbers. We show that the normal order of ω(n) is log log n for both h-free and h-full numbers. For Ω(n), the normal order is log log n over h-free numbers and h log log n over h-full numbers. We also show that ω_1(n) has normal order log log n over h-free numbers, and ω_h(n) has normal order log log n over h-full numbers. Moreover, we prove that the functions ω_k(n) with 1 < k < h do not have a normal order over h-free numbers, and that the functions ω_k(n) with k > h do not have a normal order over h-full numbers.

In their seminal work, Erdős and Kac showed that ω(n) is normally distributed over the natural numbers. Later, Liu extended this result by proving a subset generalization of the Erdős–Kac theorem. In this thesis, we leverage Liu’s framework to establish the Erdős–Kac theorem for both h-free and h-full numbers. Additionally, we show that ω_1(n) satisfies the Erdős–Kac theorem over h-free numbers, while ω_h(n) satisfies it over h-full numbers.