Average order of an arithmetic function: Difference between revisions

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==Examples==

* The average order of ''d''(''n''), the number of divisors of ''n'', is log(''n'');

* The average order of &sigma;(''n''), the sum of divisors of ''n'', is &pi;<mathsup> \frac{\pi^2}{6} n</mathsup> / 6;

* The average order of &phi;(''n'')), [[Euler's totient function]] of ''n'', is <math> \frac{6}{\ / &pi^;<sup>2} n</mathsup>;

* 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.