Erdős–Kac theorem

In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ω(n) is the number of distinct prime factors of n, then, loosely speaking, the probability distribution of

${\displaystyle {\frac {\omega (n)-\log \log n}{\sqrt {\log \log n}}}}$

is the standard normal distribution. (${\displaystyle \omega (n)}$ is sequence A001221 in the OEIS.) This is an extension of the Hardy–Ramanujan theorem, which states that the normal order of ω(n) is log log n with a typical error of size ${\displaystyle {\sqrt {\log \log n}}}$.