The Erdos-Kac Theorem states that the number of distinct prime factors of a large random integer, once centered and rescaled using the iterated logarithm of that integer, converges in distribution to a standard normal distribution. Named for Paul Erdos and Mark Kac, it is often described as a central limit theorem for the number of prime factors of an integer, one of the earliest and best known results linking probability theory to number theory.
Facts
StatementIf omega(n) is the number of distinct prime factors of n, then, loosely speaking, the probability distribution of omega(n) minus log log n, divided by the square root of log log n, is the standard normal distribution. 1 Classification
Statement Form Connections
Sources
1. Erdos-Kac theorem (Wikipedia)
Lead section, first sentence
states that if omega(n) is the number of distinct prime factors of n, then, loosely speaking, the probability distribution of omega(n) - log log n / sqrt(log log n) is the standard normal distribution
References section
Erdos, Paul, Kac, Mark (1940). The Gaussian Law of Errors in the Theory of Additive Number Theoretic Functions. American Journal of Mathematics.
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