Let $f$ be a multiplicative function. Show that there exists a multiplicative function $g$ such that, for all $p \in \mathcal{P}$ and all $k \in \mathbb{N}^*$,
$$g\left(p^k\right) = -\sum_{i=1}^{k} f\left(p^i\right) g\left(p^{k-i}\right)$$
and that it satisfies $f * g = \delta$.