Erdős–Kac theorem

The Erdős–Kac theorem states that the number of distinct prime factors of a uniformly chosen integer up to a large bound, centered and scaled by the logarithm of the logarithm of the bound, converges in distribution to a standard normal law.

Prime omega function: The arithmetic function counting the distinct prime divisors of an integer. This is the random quantity whose normalized values obey the theorem.
Distinct prime factor: A prime divisor counted once, regardless of its exponent in an integer's factorization. The theorem counts distinct divisors, not the total multiplicity of prime factors.
Paul Erdős: A Hungarian mathematician whose prolific work shaped twentieth-century number theory and combinatorics. Erdős co-authored the theorem establishing the normal law for prime-factor counts.
Additive function: An arithmetic function whose value on coprime products is the sum of its values on the factors. Prime-factor counts exemplify additive arithmetic statistics that admit normal-limit theorems.
Central limit theorem: A family of results showing that normalized sums of many weakly dependent variables approach a normal distribution. The theorem obtains a normal limit from prime-divisibility indicators resembling a sum of rare events.
Natural density: The limiting proportion of positive integers in a set among all integers up to a growing bound. Uniform sampling up to x turns the theorem into an asymptotic statement about proportions of integers.
Mark Kac: A Polish-American mathematician known for work in probability, mathematical physics, and number theory. Kac brought probabilistic methods to the study of the distribution of prime factors.
Strong law of large numbers: A theorem ensuring sample averages converge almost surely to their expected value under suitable conditions. The Erdős–Kac theorem describes fluctuations in distribution, rather than almost-sure convergence of averages.
Prime number theorem: The theorem that the number of primes up to x is asymptotic to x divided by its logarithm. Prime-counting estimates help determine the mean and variance scale, both about log log x.
Convergence in distribution: Convergence of probability laws, characterized by convergence of cumulative distribution functions at continuity points. The theorem asserts convergence of normalized counts as random variables, not pointwise convergence for each integer.

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