Large Values of for Th Order Characters and Applications to Character Sums

For any given integer we prove the existence of infinitely many and characters of order such that . We believe this bound to be the best possible. When the order is even, we obtain similar results for and , where is restricted to even (or odd) characters of order and is a fixed quadratic character. As an application of these results, we exhibit large even-order character sums, which are likely to be optimal.