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.

A more precise version states that if ψ(n) tends to infinity then as n tends to infinity

for all but an infinitesimal proportion of integers.

[edit] References

• Hardy, G.H. & Ramanujan, S. (1917), “The normal number of prime factors of a number”, Quart. J. Math. 48: 76–92 
• Hildebrand, A. (2001), “Hardy–Ramanujan theorem”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104