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

TL;DR

This paper demonstrates the existence of large values of Dirichlet L-functions within the critical strip for large moduli, matching predictions previously known only for the Riemann zeta function and quadratic L-functions under GRH.

Contribution

It extends the resonance method to show large values of Dirichlet L-functions inside the critical strip, confirming conjectured growth rates for these functions.

Findings

Existence of large L-values for large q and 1/2<σ<1

Matching the predicted growth rates for these values

Extension of the resonance method to Dirichlet L-functions

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 $\log \vert L(\sigma,\chi)\vert \gg c(\sigma)(\log q)^{1-\sigma}(\log\log q)^{-\sigma}$. 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.