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| In mathematics, the '''Hardy–Ramanujan theorem''', proved by {{harvtxt|Hardy|Ramanujan|1917}}, states that the [[Normal order of an arithmetic function|normal order]] of the number ω(''n'') of distinct [[prime factor]]s of a number ''n'' is log(log(''n'')). Roughly speaking, this means that most numbers have about this number of distinct prime factors.
| | Hello, I'm Marianne, a 28 year old from North Mundham, United Kingdom.<br>My hobbies include (but are not limited to) Worldbuilding, Creative writing and watching How I Met Your Mother.<br><br>my web-site bathroom renovation ideas ([http://www.homeimprovementdaily.com http://www.homeimprovementdaily.com]) |
| A more precise version states that for any real-valued function ψ(''n'') that tends to infinity as ''n'' tends to infinity
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| :<math>|\omega(n)-\log(\log(n))|<\psi(n)\sqrt{\log(\log(n))}</math>
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| or more traditionally
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| :<math>|\omega(n)-\log(\log(n))|<{(\log(\log(n)))}^{\frac12 +\varepsilon}</math>
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| 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.
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| A simple proof to the result {{harvtxt|Turán|1934}} was given by [[Pál Turán]], who proved that
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| :<math>\sum_{n \le x} | \omega(n) - \log\log n|^2 \ll x \log\log x \ . </math>
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| The same results are true of Ω(''n''), the number of prime factors of ''n'' counted with [[Multiplicity_(mathematics)#Multiplicity_of_a_prime_factor|multiplicity]].
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| This theorem is generalized by the [[Erdős–Kac theorem]], which shows that ω(''n'') is essentially [[Normal distribution|normally distributed]].
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| ==References==
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| *{{citation|first=G. H.|last= Hardy| authorlink=G. H. Hardy| first2= S.|last2=Ramanujan| authorlink2=Srinivasa Ramanujan |title=The normal number of prime factors of a number ''n''| journal= Quarterly Journal of Mathematics |volume= 48 |year=1917|pages= 76–92 | url=http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper35/page1.htm | jfm=46.0262.03 }}
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| * {{citation | last1=Kuo | first1=Wentang | last2=Liu | first2=Yu-Ru | chapter=The Erdős–Kac theorem and its generalizations | pages=209-216 | editor1-last=De Koninck | editor1-first=Jean-Marie | editor1-link=Jean-Marie De Koninck | editor2-last=Granville | editor2-first=Andrew | editor2-link=Andrew Granville | editor3-last=Luca | editor3-first=Florian | title=Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006 | location=Providence, RI | publisher=[[American Mathematical Society]] | series=CRM Proceedings and Lecture Notes | volume=46 | year=2008 | isbn=978-0-8218-4406-9 | zbl=1187.11024 }}
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| * {{citation | last=Turán | first=Pál | authorlink=Pál Turán | title=On a theorem of Hardy and Ramanujan | journal=Journal of the London Mathematical Society | volume=9 | pages=274-276 | year=1934 | issn=0024-6107 | zbl=0010.10401 }}
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| *{{springer|id=H/h110080|first=A.|last= Hildebrand}}
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| {{DEFAULTSORT:Hardy-Ramanujan theorem}}
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| [[Category:Analytic number theory]]
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| [[Category:Theorems about prime numbers]]
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Hello, I'm Marianne, a 28 year old from North Mundham, United Kingdom.
My hobbies include (but are not limited to) Worldbuilding, Creative writing and watching How I Met Your Mother.
my web-site bathroom renovation ideas (http://www.homeimprovementdaily.com)