Large values of quadratic Dirichlet $L$-functions near the central point

TL;DR

This paper studies the conditional large values of quadratic Dirichlet L-functions near the central point, showing they can reach magnitudes comparable to their values at the central point under certain conditions.

Contribution

It demonstrates that quadratic Dirichlet L-functions can attain large values near the central point, extending understanding of their behavior in this critical region.

Findings

L-functions exhibit large values near the central point under certain conditions

Large values near the central point are of similar magnitude to those at the central point

Conditional results depend on assumptions related to the L-functions' behavior

Abstract

In this paper, we investigate the conditional large values of the quadratic Dirichlet $L$-functions near the central point $s=1/2$. When $\sigma $ closes to $1/2$ within a suitable range, we show that $L(\sigma, \chi_d)$ have the conditional large values of the similar order of magnitude as $L(1/2, \chi_d)$.