论文标题
大二次字符总和和应用的分布
The distribution of large quadratic character sums and applications
论文作者
论文摘要
在本文中，我们研究了基本判别物$ | d | \ leq x $的原始二次字符的最大字符总和分布。特别是，我们的工作改善了蒙哥马利和沃恩的结果，并提供了有力的证据表明，贝特曼和乔瓦拉的欧米茄结果是二次性格总和是最佳的。我们还获得了具有高达$ x $的Prime判别物的真实角色的类似结果，并推断出有趣的结果，几乎所有具有大型Legendre符号总和的总优点都与$ 3 $ MODULO $ 4 $相一致。我们的结果是由Bober，Goldmakher，Granville和Koukoulopoulos的最近作品的动机，他们证明了非首席角色家族Modulo的家族相似的结果。但是，他们的方法似乎并没有推广到其他Dirichlet角色家庭。取而代之的是，我们使用一种不同，更精简的方法，该方法主要依赖于二次大筛。作为一个应用，我们考虑了关于勒文德尔符号总和的阳性的蒙哥马利问题。
In this paper, we investigate the distribution of the maximum of character sums over the family of primitive quadratic characters attached to fundamental discriminants $|d|\leq x$. 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 non-principal 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.
