Hardy–Ramanujan theorem

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

|\omega(n)-\log(\log(n))|<\psi(n)\sqrt{\log(\log(n))}

or more traditionally

|\omega(n)-\log(\log(n))|<{(\log(\log(n)))}^{\frac12 %2B\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 was given by Pál Turán.

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 in fact that distinct prime factors are essentially normally distributed.

References