> [!NOTE] Definition > A totally (or completely) multiplicative function is an [[Arithmetic Function|arithmetic function]], $f,$ such that for all $a,b\in \mathbb{N}^{+},$ $f(ab)=f(a)f(b)$ See [[Multiplicative Functions]]. # Properties # Applications ###### Examples: - [[Identity function is the only strictly increasing totally multiplicative function with f(2)=2]]. - [[Non-existence of strictly increasing totally multiplicative function with f(2)=3]].