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Erdős arcsine law

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Theorem on prime factors

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 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

2 π arcsin ⁡ u . {\displaystyle {\frac {2}{\pi }}\arcsin {\sqrt {u}}.}

References

  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.
  • 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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