# Average order of an arithmetic function  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

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

• The average order of d(n), the number of divisors of n, is log(n);
• The average order of σ(n), the sum of divisors of n, is ;
• The average order of φ(n)), Euler's totient function of n, is ;
• 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.