Average order of an arithmetic function

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 &sigma;(n), the sum of divisors of n, is $$ \frac{\pi^2}{6} n$$;
 * The average order of &phi;(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 &pi; ;
 * The Prime Number Theorem is equivalent to the statement that the von Mangoldt function &Lambda;(n) has average order 1.