Theorem

Erdos-Kac Theorem

Number Theory

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

Statement

If 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

Proof Year

1940 1

Classification

Statement Form

Existence Theorem 1

Connections

Has Statement Form

Existence Theorem

In Branch

Number Theory

Source Erdos-Kac theorem (Wikipedia)

Principal Result Of

Paul Erdos

Source Erdos-Kac theorem (Wikipedia)

Proved By

Paul Erdos

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.
Principal Result Of: Paul Erdos, Lead sentence of the article on the result
In number theory, the Erdős-Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ω(n) is the number of distinct prime factors of n, then, loosely speaking, the probability distribution of ω ( n ) − log ⁡ log ⁡ n log ⁡ log ⁡ n {\displaystyle {\frac {\omega (n)-\log \log n}{\sqrt {\log \log n}}}} is the standard normal distribution. ( ω ( n ) {\displaystyle \omega (n)} is sequence A001221 in the OEIS.)
In Branch: Number Theory, Lead sentence

Erdos-Kac Theorem

Has Statement Form

In Branch

Principal Result Of

Proved By

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