Large values of $L(1,\chi )$ for $k$-th order characters $\chi$ and applications to character sums

For any given integer $k\ge 2$ we prove the existence of infinitely many $q$ and characters $\chi (\mathrm{m}\mathrm{o}\mathrm{d}q)$ of order $k$, such that $|L(1,\chi)|\geq (e^{\gamma}+o(1))\log\log q$. We believe this bound to be best possible. When the order $k$ is even, we obtain similar results for $L(1,\chi )$ and $L(1,\chi \xi )$ where $\chi$ is restricted to even (or odd) characters of order $k$, and $\xi$ is a fixed quadratic character. As an application of these results, we exhibit large even order character sums, which are likely to be optimal.