Erdős arcsine law

In number theory, the Erdős arcsine law, named after Paul Erdős in 1969, states that the prime divisors of a number have a distribution related to the arcsine distribution.

Specifically, say that the $j$th prime factor $p$ of a given number $n$ (in the sorted sequence of distinct prime factors) is "small" when $log(log(p)) < j$. Then, for any fixed parameter $u$, in the limit as $x$ goes to infinity, the proportion of the integers $n$ less than $x$ that have fewer than $u log(log(n))$ small prime factors converges to


 * $$ \frac{2}{\pi} \arcsin{ \sqrt{u} }. $$