Large values of quadratic Dirichlet $L$-functions

TL;DR

Under GRH, this paper uses the long resonator method to establish lower bounds on the maximum size of quadratic Dirichlet L-functions across a range of the critical strip, improving previous results at the central point.

Contribution

It introduces an advanced resonator technique to derive new lower bounds for quadratic Dirichlet L-functions, extending the understanding of their maximum values within the critical strip.

Findings

Improved lower bounds for L(σ, χ_d) for σ in [1/2, 1]

Enhanced results at the central point compared to prior work

Proportion bounds for fundamental discriminants with large L-values

Abstract

Assuming the Generalized Riemann Hypothesis (GRH), we utilize the long resonator method to derive $\Omega$-results for the family of quadratic Dirichlet $L$-functions $L(\sigma, \chi_d)$, where $d$ runs over all fundamental discriminants with $|d| \leq X$ and $\sigma\in [1/2, 1]$ is fixed. This study advances understanding of the maximum size of $L(\sigma, \chi_d)$ within the segment $\sigma\in [1/2, 1]$. In particular, we improve upon Soundararajan's results at the central point and provide a lower bound on the proportion of fundamental discriminants, uniformly within an expected order of magnitude, up to optimal values of the constant for a fixed $\sigma \in (1/2, 1]$.