Erdős arcsine law

Template:Short description 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 Template:Mvar^th prime factor Template:Mvar of a given number Template:Mvar (in the sorted sequence of distinct prime factors) is "small" when Template:Math. Then, for any fixed parameter Template:Mvar, in the limit as Template:Mvar goes to infinity, the proportion of the integers Template:Mvar less than Template:Mvar that have fewer than Template:Math small prime factors converges to

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

References

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