In mathematics, specifically in number theory, the extremal orders of an arithmetic function are best possible bounds of the given arithmetic function.
Specifically, if f(n) is an arithmetic function and m(n) is a non-decreasing function that is ultimately positive and we say that m is a minimal order for f. Similarly if M(n) is a non-decreasing function that is ultimately positive and we say that M is a maximal order for f.[1]: 80 Here,
lim inf
denote the limit inferior and limit superior, respectively.
The subject was first studied systematically by Ramanujan starting in 1915.