= Completely multiplicative function
{title2=$f(1)=1,\quad f(mn)=f(m)f(n)\ \text{for all }m,n>0$}
{wiki}
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>: $\mathbf1(mn)=1=1\cdot1$ and $\operatorname{id}(mn)=mn=\operatorname{id}(m)\operatorname{id}(n)$. The <divisor function> fails this stronger condition at $m=n=2$.
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