We investigate the distribution of the maximum of character sums over the family of primitive quadratic characters attached to fundamental discriminants $\left|d\right|\le x$. In particular, our work improves results of Montgomery and Vaughan, and gives strong evidence that the Omega result of Bateman and Chowla for quadratic character sums is optimal. We also obtain similar results for real characters with prime discriminants up to $x$, and deduce the interesting consequence that almost all primes with large Legendre symbol sums are congruent to $3$ modulo $4$. Our results are motivated by a recent work of Bober, Goldmakher, Granville and Koukoulopoulos, who proved similar results for the family of nonprincipal characters modulo a large prime. However, their method does not seem to generalize to other families of Dirichlet characters. Instead, we use a different and more streamlined approach, which relies mainly on the quadratic large sieve. As an application, we consider a question of Montgomery concerning the positivity of sums of Legendre symbols.

我们研究了字符和的 马 $x$ imum 在附加到基本判别式 $\left|d\right|\le x$ 的原始二次字符族上的分布。特别是，我们的工作改进了 Montgomery 和 Vaughan 的结果，并提供了强有力的证据，证明 Bateman 和 Chowla 的二次字符和的 Omega 结果是最佳的。对于质数判别式最大为 x 的实数字符，我们还获得了类似的结果，并推导出了一个有趣的结果，即几乎所有具有大勒让德符号和的素数都与 $3$ 模全等 $4$ 。我们的结果受到 Bober、Goldmakher、Granville 和 Koukoulopoulos 最近的工作的影响，他们证明了对大素数模的非主性状家族也有类似的结果。然而，他们的方法似乎并不适用于其他狄利克雷角色家族。相反，我们使用了一种不同的、更简化的方法，它主要依赖于二次大筛子。作为一个应用，我们考虑了 Montgomery 关于勒让德符号和的正性的问题。