Completely multiplicative function ($f(1)=1,f(mn)=f(m)f(n)\text{for all }m,n>0$)

An arithmetic function $f$ is a completely multiplicative function when $f(1)=1$ and $f(mn)=f(m)f(n)$ for every pair of positive integers $m,n$. In particular $f(p^a)=f(p{)}^a$ for every prime number $p$ and every nonnegative exponent $a$. Both the constant-one arithmetic function and the identity arithmetic function are completely multiplicative functions: $1(mn)=1=1\cdot 1$ and $\mathrm{id}(mn)=mn=\mathrm{id}(m)\mathrm{id}(n)$. The divisor function fails this stronger condition at $m=n=2$.