Erdős–Kac theorem

In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, states that if &omega;(n) is the number of distinct prime factors of n, then for any fixed $$a < b$$,


 * $$\lim_{x \rightarrow \infty} \frac {1}{x} \left | \left\{ n \leq x : a \le \frac{\omega(n) - \ln \ln x}{\sqrt{\ln \ln x}} \le b \right\} \right | = \int_a^b \varphi(u)\,du $$

where


 * $$\varphi(u) = \frac{1}{\sqrt{2\pi}} e^{-u^2/2}$$

is the probability density function of the standard normal distribution, which occurs frequently in probability theory and statistics.

The theorem may be thought of as follows: choose a random member n of the set {1, 2, 3, ..., N}, all members being equally probable. Then the number &omega;(n) of distinct prime factors of n is a random variable. Its probability distribution is approximately a normal distribution whose expected value and variance are both equal to ln ln N.