Average order of an arithmetic function: Difference between revisions

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==Examples==

* The average order of ''d''(''n''), the [[Divisor function|number of divisors]] of ''n'', is log(''n'');

* The average order of σσ(''n''), the sum of divisors of ''n'', is ''n''ππ² / 6;

* The average order of φφ(''n''), [[Euler's totient function]] of ''n'', is 6''n'' / ππ²;

* The average order of ''r''(''n''), the number of ways of expressing ''n'' as a [[sum of two squares]], is ππ;

* The average order of ωω(''n''), the number of distinct [[prime factor]]s of ''n'', is log log ''n'';

* The average order of ΩΩ(''n''), the number of [[prime factor]]s of ''n'', is log log ''n'';

* The [[prime number theorem]] is equivalent to the statement that the [[von Mangoldt function]] ΛΛ(''n'') has average order 1.

==See also==

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==References==

* {{cite book | author=G.H. Hardy | authorlink=G. H. Hardy | coauthors=E.M. Wright | title=An Introduction to the Theory of Numbers | edition=6th ed. | publisher=[[Oxford University Press]] | pages=347-360347–360 | year=2008 | isbn=0-19-921986-5 }}

* {{cite book | title=Introduction to Analytic and Probabilistic Number Theory | author=Gérald Tenenbaum | series=Cambridge studies in advanced mathematics | volume=46 | publisher=[[Cambridge University Press]] | pages=36-5536–55 | year=1995 | isbn=0-521-41261-7 }}

[[Category:Arithmetic functions]]