Average order of an arithmetic function: Difference between revisions

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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 thusso 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''π² / 6;

* An average order of φ(''n''), [[Euler's totient function]] of ''n'', is 6''n'' / π²;

* An average order of ''r''(''n''), the number of ways of expressing ''n'' as a sum of two squares, is π''n''

* The average order of representations of a natural number as a sum of three squares is 4πn/3

* The average number of decompositions of a natural number into a sum of one or more consecutive prime numbers is ''nlog2''.

* 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]].

==Calculating mean values using Dirichlet series==

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In particular, the density of the [[square-free integers]] is <math>\zeta(2)^{-1}=\frac{6}{\pi^{2}}</math>.

==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''π² / 6;

* An average order of φ(''n''), [[Euler's totient function]] of ''n'', is 6''n'' / π²;

* An average order of ''r''(''n''), the number of ways of expressing ''n'' as a sum of two squares, is π''n''

* The average order of representations of a natural number as a sum of three squares is 4πn/3

* The average number of decompositions of a natural number into a sum of one or more consecutive prime numbers is ''nlog2''.

* 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]].

===Visibility of lattice points===