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

Abstract

For any given integer $k\ge 2$ we prove the existence of infinitely many $q$ and characters $\chi (\mathrm{mod}q)$ of order $k$, such that $|L(1,\chi )|\ge (e^{\gamma }+o(1))\log \log q$. 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.

18 pages

References in corpus (2)

• The frequency and the structure of large character sums
• Averages of character sums

Cited by in corpus (2)

• Extreme residues of Dedekind zeta functions
• A Strengthening of Theorems of Halász and Wirsing