Let $E$ be a $\mathbb{C}$-vector space of dimension $n \geq 1$. Let $u$ be a diagonalizable endomorphism of $E$. Show that $u$ is a permutation endomorphism if and only if there exist natural integers $c_1, \ldots, c_n$ such that, for all $k \in \mathbb{N}$,
$$\operatorname{Tr}\left(u^k\right) = \sum_{\substack{\ell=1 \\ \ell \mid k}}^{n} \ell c_\ell$$
(We sum over the values of $\ell$ dividing $k$ and belonging to $\llbracket 1, n \rrbracket$.)