Erdős–Kac theorem

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

is the standard normal distribution. This is an extension of the Hardy–Ramanujan theorem, which states that the normal order of ω(n) is log log n with a typical error of size ${\displaystyle {\sqrt {\log \log n}}}$.

For any fixed a < b,

where ${\displaystyle \Phi (a,b)}$ is the normal (or "Gaussian") distribution, defined as

More generally, if f(n) is a strongly additive function (${\displaystyle \scriptstyle f(p_{1}^{a_{1}}\cdots p_{k}^{a_{k}})=f(p_{1})+\cdots +f(p_{k})}$) with ${\displaystyle \scriptstyle |f(p)|\leq 1}$ for all prime p, then

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