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Normal order of an arithmetic function
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<h2 id="examples">Examples</h2> <ul><li>The <a href="/facts/Hardy%25E2%2580%2593Ramanujan_theorem/FNVEacAr">Hardy–Ramanujan theorem</a>: the normal order of ω(<i>n</i>), the number of distinct <a href="/facts/Prime_factor/L20FLkgn">prime factors</a> of <i>n</i>, is log(log(<i>n</i>));</li> <li>The normal order of Ω(<i>n</i>), the number of prime factors of <i>n</i> counted with <a href="/facts/Multiplicity_(mathematics)/dndUyetA">multiplicity</a>, is log(log(<i>n</i>));</li> <li>The normal order of log(<i>d</i>(<i>n</i>)), where <i>d</i>(<i>n</i>) is the number of divisors of <i>n</i>, is log(2) log(log(<i>n</i>)).</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Average_order_of_an_arithmetic_function/yf0IfE84">Average order of an arithmetic function</a></li> <li><a href="/facts/Divisor_function/rzvm1jw2">Divisor function</a></li> <li><a href="/facts/Extremal_orders_of_an_arithmetic_function/mD6uWhAa">Extremal orders of an arithmetic function</a></li> <li><a href="/facts/Tur%25C3%25A1n%25E2%2580%2593Kubilius_inequality/oWVt5DL0">Turán–Kubilius inequality</a></li></ul> <ul><li><a href="/facts/G._H._Hardy/zBkDLb05">Hardy, G.H.</a>; <a href="/facts/S._Ramanujan/HBPr4bzd">Ramanujan, S.</a> (1917). <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper35/page1.htm">"The normal number of prime factors of a number <i>n</i>"</a>. <i>Quart. J. Math</i>. 48: 76–92. <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:46.0262.03">46.0262.03</a>.</li> <li><a href="/facts/G._H._Hardy/zBkDLb05">Hardy, G. H.</a>; <a href="/facts/E._M._Wright/SS01ywc2">Wright, E. M.</a> (2008) [1938]. <i><a href="/facts/An_Introduction_to_the_Theory_of_Numbers/YNc01PUg">An Introduction to the Theory of Numbers</a></i>. Revised by <a href="/facts/Roger_Heath-Brown/uMHxGtbd">D. R. Heath-Brown</a> and <a href="/facts/Joseph_H._Silverman/acyPld7K">J. H. Silverman</a>. Foreword by <a href="/facts/Andrew_Wiles/bdc3k3pQ">Andrew Wiles</a>. (6th ed.). Oxford: <a href="/facts/Oxford_University_Press/CYyTzzWG">Oxford University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-921986-5. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243">2445243</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1159.11001">1159.11001</a>.. p. 473</li> <li>Sándor, Jozsef; Crstici, Borislav (2004), <i>Handbook of number theory II</i>, Dordrecht: Kluwer Academic, p. 332, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-4020-2546-7, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1079.11001">1079.11001</a></li> <li>Tenenbaum, Gérald (1995). <i>Introduction to Analytic and Probabilistic Number Theory</i>. Cambridge studies in advanced mathematics. Vol. 46. Translated from the 2nd French edition by C.B.Thomas. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. pp. 299–324. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-41261-7. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0831.11001">0831.11001</a>.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/NormalOrder.html">"Normal Order"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>