Erdös-Kac Theorem

A deeper result than the Hardy-Ramanujan Theorem. Let $N(x,a,b)$ be the number of Integers in $[3, x]$ such that inequality

$$a\leq {\omega(n)-\ln\ln n\over \sqrt{\ln\ln n}} \leq b$$

holds, where $\omega(n)$ is the number of different Prime factors of $n$. Then

$$\lim_{x\to\infty} N(x,a,b)={(x+o(x))\over\sqrt{2\pi}} \int_a^b e^{-t^2/2}\,dt.$$

The theorem is discussed in Kac (1959).

References

Kac, M. Statistical Independence in Probability, Analysis and Number Theory. New York: Wiley, 1959.

Riesel, H. ``The Erdös-Kac Theorem.'' Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 158-159, 1994.