https://de.wikipedia.org/w/api.php?action=feedcontributions&feedformat=atom&user=Spectral+sequence Wikipedia - Benutzerbeiträge [de] 2025-07-25T18:25:30Z Benutzerbeiträge MediaWiki 1.45.0-wmf.11 https://de.wikipedia.org/w/index.php?title=Normale_Gr%C3%B6%C3%9Fenordnung&diff=180608874 Normale Größenordnung 2013-07-27T20:19:40Z <p>Spectral sequence: /* References */ better</p> <hr /> <div>In [[number theory]], a '''normal order of an arithmetic function''' is some simpler or better-understood function which &quot;usually&quot; takes the same or closely approximate values.<br /> <br /> Let &amp;fnof; be a function on the [[natural number]]s. We say that ''g'' is a '''normal order''' of &amp;fnof; if for every ''&amp;epsilon;''&amp;nbsp;&gt;&amp;nbsp;0, the inequalities<br /> <br /> :&lt;math&gt; (1-\varepsilon) g(n) \le f(n) \le (1+\varepsilon) g(n) \, &lt;/math&gt;<br /> <br /> hold for ''[[almost all]]'' ''n'': that is, if the proportion of ''n'' &amp;le; ''x'' for which this does not hold tends to 0 as ''x'' tends to infinity.<br /> <br /> It is conventional to assume that the approximating function ''g'' is [[Continuous function|continuous]] and [[Monotonic function|monotone]].<br /> <br /> ==Examples==<br /> * The [[Hardy–Ramanujan theorem]]: the normal order of &amp;omega;(''n''), the number of distinct [[prime factor]]s of ''n'', is log(log(''n''));<br /> * The normal order of &amp;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)&amp;nbsp;log(log(''n'')).<br /> <br /> ==See also==<br /> * [[Average order of an arithmetic function]]<br /> * [[Divisor function]]<br /> * [[Extremal orders of an arithmetic function]]<br /> <br /> ==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 /> <br /> ==External links==<br /> * {{MathWorld|urlname=NormalOrder|title=Normal Order}}<br /> <br /> [[Category:Arithmetic functions]]<br /> <br /> {{numtheory-stub}}</div> Spectral sequence