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Hardy-Ramanujan Theorem


Let omega(n) be the number of distinct prime factors of n. If Psi(x) tends steadily to infinity with x, then

 lnlnx-Psi(x)sqrt(lnlnx)<omega(n)<lnlnx+Psi(x)sqrt(lnlnx)

for almost all numbers n<x. "almost all" means here the frequency of those integers n in the interval 1<=n<=x for which

 |omega(n)-lnlnx|>Psi(x)sqrt(lnlnx)

approaches 0 as x->infty.


See also

Distinct Prime Factors, Erdős-Kac Theorem

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Cite this as:

Weisstein, Eric W. "Hardy-Ramanujan Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Hardy-RamanujanTheorem.html

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