Hardy–Ramanujan theorem

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In mathematics, the Hardy–Ramanujan theorem, proved by G. H. Hardy and Srinivasa 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

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


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.


  • Hardy, G. H.; Ramanujan, S. (1917), "The normal number of prime factors of a number n", Quarterly Journal of Mathematics, 48: 76–92, JFM 46.0262.03
  • Kuo, Wentang; Liu, Yu-Ru (2008), "The Erdős–Kac theorem and its generalizations", in De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian, Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006, CRM Proceedings and Lecture Notes, 46, Providence, RI: American Mathematical Society, pp. 209–216, ISBN 978-0-8218-4406-9, Zbl 1187.11024
  • Turán, Pál (1934), "On a theorem of Hardy and Ramanujan", Journal of the London Mathematical Society, 9: 274–276, doi:10.1112/jlms/s1-9.4.274, ISSN 0024-6107, Zbl 0010.10401
  • Hildebrand, A. (2001) [1994], "H/h110080", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4