Erdős arcsine law

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

Specifically, say that the jth 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

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

References

• Manstavičius, E. (1994), "A proof of the Erdős arcsine law", Probability theory and mathematical statistics (Vilnius, 1993), Vilnius: TEV, pp. 533–539, MR 1649597