Extremal orders of an arithmetic function

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

The subject was first studied systematically by Ramanujan starting in 1915.[1]:87