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In [[number theory]], an '''average order of an arithmetic function''' is some simpler or better-understood function which takes the same values "on average".
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Let ''f'' be an [[arithmetic function]]. We say that an ''average order'' of ''f'' is ''g'' if
 
:<math> \sum_{n \le x} f(n) \sim \sum_{n \le x} g(n) </math>
 
as ''x'' tends to infinity.
 
It is conventional to choose an approximating function ''g'' that is [[Continuous function|continuous]] and [[Monotonic function|monotone]]. But even thus an average order is of course not unique.  
 
==Examples==
* An average order of ''d''(''n''), the [[Divisor function|number of divisors]] of ''n'', is log(''n'');
* An average order of σ(''n''), the sum of divisors of ''n'', is ''n''π<sup>2</sup> / 6;
* An average order of φ(''n''), [[Euler's totient function]] of ''n'', is 6''n'' / π<sup>2</sup>;
* An average order of ''r''(''n''), the number of ways of expressing ''n'' as a sum of two squares, is π;
* An average order of ω(''n''), the number of distinct [[prime factor]]s of ''n'', is log log ''n'';
* An average order of Ω(''n''), the number of prime factors of ''n'', is log log ''n'';
* The [[prime number theorem]] is equivalent to the statement that the [[von Mangoldt function]] Λ(''n'') has average order 1;
* An average order of μ(''n''), the [[Möbius function]], is zero; this is again equivalent to the [[prime number theorem]].
 
==Better average order==
 
This notion is best discussed through an example. From
:<math> \sum_{n\le x}d(n)=x\log x+(2\gamma-1)x+o(x)</math>
(<math>\gamma</math> is the [[Euler-Mascheroni constant]]) and
:<math> \sum_{n\le x}\log n=x\log x-x+O(\log x),</math>
we have the asymptotic relation
:<math>\sum_{n\le x}(d(n)-(\log n+2\gamma))=o(x)\quad(x\rightarrow\infty),</math>
which suggests that the function <math>\log n+2\gamma</math> is a better choice of average order for <math>d(n)</math> than simply <math>\log n</math>.  
 
 
==See also==
* [[Divisor summatory function]]
* [[Normal order of an arithmetic function]]
* [[Extremal orders of an arithmetic function]]
 
==References==
* {{Hardy and Wright|citation=cite book}}  Pp.347–360
* {{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]] | year=1995 | isbn=0-521-41261-7 | zbl=0831.11001 | pages=36–55 }}
 
[[Category:Arithmetic functions]]
 
{{numtheory-stub}}

Latest revision as of 21:30, 25 October 2014

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