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 (sequence A001221 in the OEIS), then, loosely speaking, the probability distribution ofdigits in nof distinct primesdeviation 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 (sequence A001221 in the OEIS), 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 log ⁡ log ⁡ n {displaystyle {sqrt {log log n}}} . For any fixed a < b, where Φ ( a , b ) {displaystyle Phi (a,b)} is the normal (or 'Gaussian') distribution, defined as More generally, if f(n) is a strongly additive function ( f ( p 1 a 1 ⋯ p k a k ) = f ( p 1 ) + ⋯ + f ( p k ) {displaystyle scriptstyle f(p_{1}^{a_{1}}cdots p_{k}^{a_{k}})=f(p_{1})+cdots +f(p_{k})} ) with | f ( p ) | ≤ 1 {displaystyle scriptstyle |f(p)|leq 1} for all prime p, then