# Normal order of an arithmetic function

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

A function, perhaps simpler or better-understood, which "usually" takes the same or closely approximate values as a given arithmetic function.

Let $f$ be a function on the natural numbers. We say that the normal order of $f$ is $g$ if for every $\epsilon > 0$, the inequalities $$(1-\epsilon) g(n) \le f(n) \le (1+\epsilon) g(n)$$ hold for almost all $n$: that is, the proportion of $n < x$ for which this does not hold tends to 0 as $x$ tends to infinity.

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