Let $f$ and $g$ be two multiplicative functions. Show that if $$\forall p \in \mathcal{P}, \quad \forall k \in \mathbb{N}^*, \quad f\left(p^k\right) = g\left(p^k\right)$$ then $f = g$.
Let $f$ and $g$ be two multiplicative functions. Show that if
$$\forall p \in \mathcal{P}, \quad \forall k \in \mathbb{N}^*, \quad f\left(p^k\right) = g\left(p^k\right)$$
then $f = g$.