# Average order of an arithmetic function

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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 $\sum_{n \le x} f(n) \sim \sum_{n \le x} g(n)$

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 $\frac{\pi^2}{6} n$;
• The average order of φ(n)), Euler's totient function of n, is $\frac{6}{\pi^2} n$;
• 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.