Erdős–Kac theorem
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In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, states that if ω(x) is the number of distinct prime factors of x, then
where
is the probability density function of the standard normal distribution, which occurs incessantly in probability theory and statistics.
The theorem may be thought of as follows: choose a random member x of the set {1, 2, 3, ..., n}, all members being equally probable. Then the number ω(x) of distinct prime factors of x is a random variable. Its probability distribution is approximately a normal distribution whose expected value and variance are both equal to ln ln n. The approximation can be made as close as desired by making n big enough.
[edit] References
- Paul Erdős and Mark Kac, "The Gaussian Law of Errors in the Theory of Additive Number Theoretic Functions", American Journal of Mathematics, volume 62, No. 1/4, (1940), pages 738—742.
