# 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}