Hardy–Ramanujan theorem
In mathematics, the Hardy–Ramanujan theorem, proved by, states that the normal order of the number ω of distinct prime factors of a number n is log.
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 that tends to infinity as n tends to infinity
or more traditionally
for almost all integers. That is, let g be the number of positive integers n less than x for which the above inequality fails: then g/x converges to zero as x goes to infinity.History
A simple proof to the result was given by Pál Turán, who used the Turán sieve to prove thatGeneralizations
The same results are true of Ω, the number of prime factors of n counted with multiplicity.
This theorem is generalized by the Erdős–Kac theorem, which shows that ω is essentially normally distributed.
