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

${\displaystyle \sum _{n\leq x}f(n)\sim \sum _{n\leq x}g(n)}$

as x tends to infinity.

It is conventional to assume that the approximating function g is continuous and monotone.