Hardy–Ramanujan theorem

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In mathematics, the Hardy–Ramanujan theorem, proved by Hardy & Ramanujan (1917), states that the normal order of the number ω(n) of distinct prime factors of a number n is log(log(n)).

Roughly speaking, this means that most numbers have about this number of distinct prime factors.

Precise statement[edit]

A more precise version states that for any real-valued function ψ(n) that tends to infinity as n tends to infinity


or more traditionally

|\omega(n)-\log(\log(n))|<{(\log(\log(n)))}^{\frac12 +\varepsilon}

for almost all (all but an infinitesimal proportion of) integers. That is, let g(x) be the number of positive integers n less than x for which the above inequality fails: then g(x)/x converges to zero as x goes to infinity.


A simple proof to the result Turán (1934) was given by Pál Turán, who used the Turán sieve to prove that

\sum_{n \le x} | \omega(n) - \log\log n|^2 \ll x \log\log x \ .


The same results are true of Ω(n), the number of prime factors of n counted with multiplicity. This theorem is generalized by the Erdős–Kac theorem, which shows that ω(n) is essentially normally distributed.