Average order of an arithmetic function: Difference between revisions

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In [[mathematics]], in the field of [[number theory]], the '''average order of an arithmetic function''' is some simpler or better-understood function which takes the same values "on average".
In [[mathematics]], in the field of [[number theory]], the '''average order of an arithmetic function''' is some simpler or better-understood function which takes the same values "on average".


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==Examples==
==Examples==
* The average order of ''d''(''n''), the number of divisors of ''n'', is log(''n'');
* The average order of ''d''(''n''), the [[number of divisors function|number of divisors]] of ''n'', is log(''n'');
* The average order of &sigma;(''n''), the sum of divisors of ''n'', is <math> \frac{\pi^2}{6} n</math>;
* The average order of &sigma;(''n''), the [[Sum-of-divisors function|sum of divisors]] of ''n'', is <math> \frac{\pi^2}{6} n</math>;
* The average order of &phi;(''n'')), [[Euler's totient function]] of ''n'', is <math> \frac{6}{\pi^2} n</math>;
* The average order of &phi;(''n'')), [[Euler's totient function]] of ''n'', is <math> \frac{6}{\pi^2} n</math>;
* The average order of ''r''(''n'')), the number of ways of expressing ''n'' as a [[sum of two squares]], is &pi; ;
* The average order of ''r''(''n'')), the number of ways of expressing ''n'' as a [[sum of two squares]], is &pi; ;
* The [[Prime Number Theorem]] is equivalent to the statement that the [[von Mangoldt function]] &Lambda;(''n'') has average order 1.
* The [[Prime Number Theorem]] is equivalent to the statement that the [[von Mangoldt function]] &Lambda;(''n'') has average order 1.
==See also==
* [[Divisor function]]
* [[Normal order of an arithmetic function]]


==References==
==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=347-360 | year=2008 | isbn=0-19-921986-5 }}  
* {{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=347-360 | 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=36-55 | year=1995 | isbn=0-521-41261-7 }}
* {{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=36-55 | year=1995 | isbn=0-521-41261-7 }}[[Category:Suggestion Bot Tag]]
 
[[Category:Arithmetic functions]]
 
{{numtheory-stub}}

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In mathematics, in the field of number theory, the average order of an arithmetic function is some simpler or better-understood function which takes the same values "on average".

Let f be a function on the natural numbers. We say that the average order of f is g if

as x tends to infinity.

It is conventional to assume that the approximating function g is continuous and monotone.

Examples

References