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<div>In [[number theory]], a '''normal order of an arithmetic function''' is some simpler or better-understood function which "usually" takes the same or closely approximate values.<br />
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Let &fnof; be a function on the [[natural number]]s. We say that ''g'' is a '''normal order''' of &fnof; if for every ''&epsilon;''&nbsp;>&nbsp;0, the inequalities<br />
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:<math> (1-\varepsilon) g(n) \le f(n) \le (1+\varepsilon) g(n) \, </math><br />
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hold for ''[[almost all]]'' ''n'': that is, if the proportion of ''n'' &le; ''x'' for which this does not hold tends to 0 as ''x'' tends to infinity.<br />
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It is conventional to assume that the approximating function ''g'' is [[Continuous function|continuous]] and [[Monotonic function|monotone]].<br />
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==Examples==<br />
* The [[Hardy–Ramanujan theorem]]: the normal order of &omega;(''n''), the number of distinct [[prime factor]]s of ''n'', is log(log(''n''));<br />
* The normal order of &Omega;(''n''), the number of prime factors of ''n'' counted with [[multiplicity (mathematics)|multiplicity]], is log(log(''n''));<br />
* The normal order of log(''d''(''n'')), where ''d''(''n'') is the number of divisors of ''n'', is log(2)&nbsp;log(log(''n'')).<br />
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==See also==<br />
* [[Average order of an arithmetic function]]<br />
* [[Divisor function]]<br />
* [[Extremal orders of an arithmetic function]]<br />
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==References==<br />
* {{cite journal| first=G.H. | last=Hardy| authorlink=G. H. Hardy| coauthors=[[S. Ramanujan]]|title=The normal number of prime factors of a number ''n'' |journal= Quart. J. Math. | volume= 48 | year=1917 | pages= 76–92 | url=http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper35/page1.htm | jfm=46.0262.03 }}<br />
* {{Hardy and Wright | citation=cite book | page=473 }}. p.473<br />
* {{citation | last1=Sándor | first1=Jozsef | last2=Crstici | first2=Borislav | title=Handbook of number theory II | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=1-4020-2546-7 | page=332 | zbl=1079.11001 }}<br />
* {{cite book | title=Introduction to Analytic and Probabilistic Number Theory | first=Gérald | last=Tenenbaum | others=Translated from the 2nd French edition by C.B.Thomas | series=Cambridge studies in advanced mathematics | volume=46 | publisher=[[Cambridge University Press]] | year=1995 | isbn=0-521-41261-7 | zbl=0831.11001 | pages=299–324 }}<br />
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==External links==<br />
* {{MathWorld|urlname=NormalOrder|title=Normal Order}}<br />
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[[Category:Arithmetic functions]]<br />
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{{numtheory-stub}}</div>Spectral sequence