Note on large quadratic character sums
Zikang Dong, Yutong Song, Ruihua Wang, Shengbo Zhao
TL;DR
The paper investigates conditional large values of quadratic Dirichlet character sums under GRH and proves an Omega-type lower bound for real primitive characters in a specific regime where $x$ satisfies $\exp\left(4\sqrt{\log X\log_2 X}\log_3 X\right)\le x\le \exp((\log X)^{1/2+\varepsilon})$. Using the resonance method with a carefully designed resonator, it achieves the bound $$$\max_{X<|d|\le 2X\atop d\in\mathcal{F}}\sum_{n\le |d|/x}\chi_d(n)\ge \sqrt{\frac{X}{x}}\exp\left(\left(\frac{\sqrt{2}}{2}+o(1)\right)\sqrt{\frac{\log X}{\log_2 X}}\right)$$, demonstrating substantial large-value behavior for real characters in this range. The results extend Theorem 1.2 in prior work (lqcs) and complement parts of Hou's findings, contributing to a sharper understanding of the distribution and extremal size of quadratic character sums under GRH. This advances insights into the limits of Pólya–Vinogradov-type bounds and the structure of large character sums in the real-character setting.
Abstract
In this article, we investigate the conditional large values of quadratic Dirichlet character sums. We prove an Omega result for quadratic character sums under the assumption of the generalized Riemann hypothesis.