Average order of an arithmetic function: Difference between revisions

Revision as of 18:59, 7 August 2018

Revision as of 02:06, 22 February 2019

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exists, it is said that <math>f</math> has a '''mean value''' ('''average value''') <math>c</math>.

==Examples==

==Examples==<!-- &#x200A; is hair space (&hairsp; doesn't work). -->

* An average order of {{math|''d''&#x200A;(''n'')}}, the [[Divisor function|number of divisors]] of {{math|''n''}}, is {{math|log ''n''}};

* An average order of {{math|''σ''&#x200A;(''n'')}}, the [[Divisor function|sum of divisors]] of {{math|''n''}}, is {{math|''n''&#x200A;π²{{thinsp|/}}6}};

* An average order of {{math|''φ''&#x200A;(''n'')}}, [[Euler's totient function]] of {{math|''n''}}, is {{math|6&#x200A;''n''{{thinsp|/}}π²}};

* An average order of {{math|''r''&#x200A;(''n'')}}, the number of ways of expressing {{math|''n''}} as a sum of two squares, is {{math|π}};

* The average order of representations of a natural number as a sum of three squares is {{math|4π&#x200A;''n''{{thinsp|/}}3}};

* The average number of decompositions of a natural number into a sum of one or more consecutive prime numbers is {{math|''n'' log&#x200A;2log2}};

* An average order of {{math|''ω''&#x200A;(''n'')}}, the [[Prime factors|number of distinct prime factors]] of {{math|''n''}}, is {{math|log&#x200A;logloglog ''n''}};

* An average order of {{math|Ω&#x200A;(''n'')}}, the [[Prime factors|number of prime factors]] of {{math|''n''}}, is {{math|log&#x200A;logloglog ''n''}};

* The [[prime number theorem]] is equivalent to the statement that the [[von Mangoldt function]] {{math|Λ&#x200A;(''n'')}} has average order 1;

* An average order of {{math|''μ''&#x200A;(''n'')}}, the [[Möbius function]], is zero; this is again equivalent to the [[prime number theorem]].

==Calculating mean values using Dirichlet series==