A deeper result than the Hardy-Ramanujan theorem. Let N(x, a, b) be the number of integers in [n, x] such that inequality a<=(ω(n) - ln ln n)/sqrt(ln ln n)<=b holds, where ω(n) is the number of distinct prime factors of n. Then lim_(x->∞) N(x, a, b) | = | (x + o(x))/sqrt(2π) integral_a^b e^(-t^2/2) d t | = | (x + o(x))/2[erf(b/sqrt(2)) - erf(a/sqrt(2))], where o(x) is a Landau symbol. The theorem is discussed in Kac.

distinct prime factors | Hardy-Ramanujan theorem