Large values of quadratic Dirichlet L-functions

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 $\sigma\in [1/2, 1]$, particularly improving Soundararajan's result at the central point.