LAUSR

For any given integer $k\geqslant 2$ we prove the existence of infinitely many $q$ and characters $\unicode[STIX]{x1D712}\,(\text{mod}\;q)$ of order $k$ such that $|L(1,\unicode[STIX]{x1D712})|\geqslant (\text{e}^{\unicode[STIX]{x1D6FE}}+o(1))\log \log q$ . We believe this bound to be the best possible. When the order $k$ is even, we obtain similar results for $L(1,\unicode[STIX]{x1D712})$ and $L(1,\unicode[STIX]{x1D712}\unicode[STIX]{x1D709})$ , where $\unicode[STIX]{x1D712}$ is restricted to even (or odd) characters of order $k$ and $\unicode[STIX]{x1D709}$ is a fixed quadratic character. As an application of these results, we exhibit large even-order character sums, which are likely to be optimal.