On the proportion of characters with large values of $L(1,χ)$

Christoph Aistleitner, Kamalakshya Mahatab, Alexandre Peyrot

Published 2018-03-02Version 1

The first and second authors of this paper were recently involved in the development of a version of the resonance method which allowed to prove the existence of large values of the Riemann zeta function on the 1-line, up to a level which essentially matches the prediction for the maximal order of such large values. In the present paper we show how the method can be adapted to prove the existence of large values of $|L(1, \chi)|$. More precisely, we show that for all sufficiently large $q$ there is a non-principal character $\chi$ (mod $q$) for which $|L(1,\chi)| \geq e^\gamma \left(\log_2 q + \log_3 q - C \right)$, where $C$ is an absolute constant. We also give bounds for the proportion of characters for which $|L(1,\chi)|$ is of this order of magnitude, as a function of $C$.