The Hardy–Ramanujan Theorem on the Number of Distinct Prime Divisors

Abstract

Let ω(n) denote the number of distinct prime divisors of n; that is, $$\displaystyle{\omega (n) =\sum _{p\vert n}1.}$$ Thus, for example, ω(1) = 0, ω(2) = 1, ω(9) = 1, ω(60) = 3. The values of ω(n) obviously fluctuate wildly as n → ∞, since ω(p) = 1, for every prime p.