# Average order of an arithmetic function

2010 Mathematics Subject Classification: Primary: 11A25 [MSN][ZBL]

Some simpler or better-understood function which takes the same values "on average" as an arithmetic function.

Let $f$, $g$ be functions on the natural numbers. We say that $f$ has average order $g$ if the asymptotic equality $$\sum_{n \le x} f(n) \sim \sum_{n \le x} g(n)$$ holds as $x$ tends to infinity.

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