Multiplicative function

In number theory, a multiplicative function is an arithmetic function ${\displaystyle f}$ of a positive integer ${\displaystyle n}$ with the property that ${\displaystyle f(1)=1}$ and $${\displaystyle f(ab)=f(a)f(b)}$$ whenever ${\displaystyle a}$ and ${\displaystyle b}$ are coprime.

An arithmetic function is said to be completely multiplicative (or totally multiplicative) if ${\displaystyle f(1)=1}$ and ${\displaystyle f(ab)=f(a)f(b)}$ holds for all positive integers ${\displaystyle a}$ and ${\displaystyle b}$, even when they are not coprime.