Extremal orders of an arithmetic function

The extremal orders of an arithmetic function in number theory, a branch of mathematics, are the 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

${\displaystyle \liminf _{n\to \infty }{\frac {f(n)}{m(n)}}=1}$

we say that m is a minimal order for f. Similarly if M(n) is a non-decreasing function that is ultimately positive and

${\displaystyle \limsup _{n\to \infty }{\frac {f(n)}{M(n)}}=1}$

we say that M is a maximal order for f. Here, ${\displaystyle \liminf _{n\to \infty }}$ and ${\displaystyle \limsup _{n\to \infty }}$ denote the limit inferior and limit superior, respectively.

The subject was first studied systematically by Ramanujan starting in 1915.