Large values of Dirichlet $L$- functions inside the critical strip

Abstract

In the present paper, we study large values of Dirichlet $L$- functions inside the critical strip. For every $1/2<\sigma <1$, we show that for $q$ sufficiently large, there exists a non-principal character $\chi$ modulo $q$ and a constant $c(\sigma )>0$ such that $\log |L(\sigma ,\chi )|\gg c(\sigma )(\log q{)}^{1-\sigma }(\log \log q{)}^{-\sigma }$. This matches the believed prediction for these values which was previously known only for the Riemann zeta function since Montgomery, or conditionally on GRH for quadratic $L$- functions due to Lamzouri. In a recent work involving the author, a new implementation of the resonance method was presented in order to exhibit large values of the Riemann zeta function on the line $\Re (s)=1$. We show how to adapt the argument to our setting.

A joint version has been prepared with the article arXiv:1803.00760 in order to get a complete treatment of $L$ -functions in the strip $(1/2,1)$. Thus, the paper arXiv:1803.00760 has been updated with the current author and contains the content of the present paper