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LARGE VALUES OF $L(1,\unicode[STIX]{x1D712})$ FOR $k$TH ORDER CHARACTERS $\unicode[STIX]{x1D712}$ AND APPLICATIONS TO CHARACTER SUMS

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  26 July 2016

Youness Lamzouri
Youness Lamzouri*
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J1P3, Canada email [email protected]
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Abstract

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.

MSC classification

Primary: 11M20: Real zeros of $L(s, chi)$; results on $L(1, chi)$ 11M06: $zeta (s)$ and $L(s, chi)$
Type
Research Article
Information
Mathematika , Volume 63 , Issue 1 , 2017 , pp. 53 - 71
Copyright
Copyright © University College London 2016 

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