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| 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.
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| Let ƒ be a function on the [[natural number]]s. We say that ''g'' is a '''normal order''' of ƒ if for every ''ε'' > 0, the inequalities | |
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| :<math> (1-\varepsilon) g(n) \le f(n) \le (1+\varepsilon) g(n) \, </math>
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| hold for ''[[almost all]]'' ''n'': that is, if the proportion of ''n'' ≤ ''x'' for which this does not hold tends to 0 as ''x'' tends to infinity.
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| It is conventional to assume that the approximating function ''g'' is [[Continuous function|continuous]] and [[Monotonic function|monotone]].
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| ==Examples==
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| * The [[Hardy–Ramanujan theorem]]: the normal order of ω(''n''), the number of distinct [[prime factor]]s of ''n'', is log(log(''n''));
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| * The normal order of Ω(''n''), the number of prime factors of ''n'' counted with [[multiplicity (mathematics)|multiplicity]], is log(log(''n''));
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| * The normal order of log(''d''(''n'')), where ''d''(''n'') is the number of divisors of ''n'', is log(2) log(log(''n'')).
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| ==See also==
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| * [[Average order of an arithmetic function]]
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| * [[Divisor function]]
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| * [[Extremal orders of an arithmetic function]]
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| ==References==
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| * {{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 }}
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| * {{Hardy and Wright | citation=cite book | page=473 }}. p.473
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| * {{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 }}
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| * {{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 }}
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| ==External links==
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| * {{MathWorld|urlname=NormalOrder|title=Normal Order}}
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| [[Category:Arithmetic functions]]
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| {{numtheory-stub}}
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8 Let me first start by introducing people. My name is Moses Walt. One of the things love most is badminton and I'm going to never stop doing getting this done. My day job is a manager. For years he's been coping with Arkansas. See what's new in her website here: http://zigzigziggy.co.uk/activity/p/705262/
Check out my weblog - nowe mieszkania Warszawa Wilanów