Erdős-Kac theorem
= Erdős-Kac theorem
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= Erdős-Kac central limit theorem
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For a real <strongly additive arithmetic function> with uniformly bounded values on <primes>, put $A(x)=\sum_{p\le x}f(p)/p$ and $B(x)^2=\sum_{p\le x}f(p)^2/p$. If $B(x)\to\infty$, then $(f(n)-A(N))/B(N)$ under uniform sampling from $[N]$ converges to the <standard normal distribution>. For the <prime omega function>, the mean and variance scales are $\log\log N$.