Let ω(n) be the number of distinct prime factors of n. If Ψ(x) tends steadily to infinity with x, then ln ln x - Ψ(x) sqrt(ln ln x)<ω(n)Ψ(x) sqrt(ln ln x) approaches 0 as x->∞.

distinct prime factors | Erdős-Kac theorem

Godfrey Harold Hardy | Srinivasa Aiyangar Ramanujan