Average order of an arithmetic function: Difference between revisions

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* The average order of ''r''(''n'')), the number of ways of expressing ''n'' as a [[sum of two squares]], is π ;
* The average order of ''r''(''n'')), the number of ways of expressing ''n'' as a [[sum of two squares]], is π ;
* The [[Prime Number Theorem]] is equivalent to the statement that the [[von Mangoldt function]] Λ(''n'') has average order 1.
* The [[Prime Number Theorem]] is equivalent to the statement that the [[von Mangoldt function]] Λ(''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 }}

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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