Hardy–Ramanujan theorem

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

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

or more traditionally

${\displaystyle |\omega (n)-\log \log n|<{(\log \log n)}^{{\frac {1}{2}}+\varepsilon }}$

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 Turán (1934) was given by Pál Turán, who used the Turán sieve to prove that

${\displaystyle \sum _{n\leq x}|\omega (n)-\log \log x|^{2}\ll x\log \log x\ .}$

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 (eds.), Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006, CRM Proceedings and Lecture Notes, vol. 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 (4): 274–276, doi:10.1112/jlms/s1-9.4.274, ISSN 0024-6107, Zbl 0010.10401
• Hildebrand, A. (2001) [1994], "Hardy-Ramanujan theorem", Encyclopedia of Mathematics, EMS Press