Large values of quadratic character sums revisited
Zikang Dong, Ruihua Wang, Weijia Wang, Hao Zhang
TL;DR
The paper proves a GRH-assisted Omega-type lower bound for large quadratic character sums with summation length $x$ exceeding $\sqrt{q}$, extending dual ideas from La Bretèche–Tenenbaum to quadratic characters. It uses the resonance method to convert the problem into estimating moments of a carefully chosen resonator, guided by Polya’s Fourier expansion and a GRH-conditioned discriminant sum formula. The main result shows that for large $X$ and $\exp((\log X)^{1/2+\varepsilon})<x<X^{1/2}$, one has $\max_{X<|d|\le 2X, d\in\mathcal{F}}\sum_{n\le|d|/x}\chi_d(n) \ge \sqrt{\dfrac{X}{x}}\exp\big((1+o(1))\sqrt{\dfrac{\log(\sqrt{X}/x)\log_3(\sqrt{X}/x)}{\log_2(\sqrt{X}/x)}}\big)$. This sharpens our understanding of how large quadratic character sums can be in the regime beyond the square-root barrier and demonstrates the efficacy of the resonance method under GRH. The work complements existing BT-type results for zeta sums and extends their scope to quadratic characters, with potential implications for related extremal problems in analytic number theory.
Abstract
We study large values of quadratic character sums with summation lengths exceeding the square root of the modulus. Assuming the Generalized Riemann Hypothesis, we obtain a new Omega result.