Erdős arcsine law

In number theory, the Erdős arcsine law, named after Paul Erdős in 1969,^[1] 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}}.

References

^ Manstavičius, E. (2020-05-18). "A proof of the Erdös arcsine law". Probability Theory and Mathematical Statistics. De Gruyter. pp. 533–564. doi:10.1515/9783112319321-032. ISBN 978-3-11-231932-1.

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

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[1] Manstavičius, E. (2020-05-18). "A proof of the Erdös arcsine law". Probability Theory and Mathematical Statistics. De Gruyter. pp. 533–564. doi:10.1515/9783112319321-032. ISBN 978-3-11-231932-1.

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