Английская Википедия:Hardy–Ramanujan theorem

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In mathematics, the Hardy–Ramanujan theorem, proved by Ramanujan and checked by Hardy 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

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

<math>|\omega(n)-\log\log n|<\psi(n)\sqrt{\log\log n}</math>

or more traditionally

<math>|\omega(n)-\log\log n|<{(\log\log n)}^{\frac12 +\varepsilon}</math>

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.

History

A simple proof to the result Шаблон:Harvtxt was given by Pál Turán, who used the Turán sieve to prove that

<math>\sum_{n \le x} | \omega(n) - \log\log x|^2 \ll x \log\log x \ . </math>

Generalizations

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.

References

• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Eom