AI 中文总结
本文是一篇关于哥德巴赫问题例外集的综述，介绍了哈代-李特尔伍德圆法及相关进展，给出主弧显式公式的新结果，并发现稀疏版哈代-李特尔伍德猜想下无例外零点。
AI 中文摘要
我们研究整数表示为至多两个素数之和的例外个数的估计。本文大部分内容是一篇综述，概述现有结果。我们从传奇的哈代-李特尔伍德圆法开始，展示它如何为1975年蒙哥马利-沃恩和2018年平茨的幂节省铺平道路。最后我们给出关于主弧的完全显式公式这一新结果，另一项新发现是在稀疏版哈代-李特尔伍德猜想下不存在例外零点。本文的综述部分旨在让未接触过这些技巧的读者也能理解。
英文摘要
We study the estimates for the number of exceptions to the representation of integers as the sum of at most two prime numbers. Most of this article is a survey that gives an overview of existing results. We begin with the legendary Hardy-Littlewood circle method and show how it paved the way to a power saving by Montgomery-Vaughan in 1975 and Pintz in 2018. We conclude with a new result that is a fully explicit formula for the major arcs. Another new observation is the non-existence of exceptional zeros under a sparse version of the Hardy-Littlewood conjecture. The survey part of this article aims to be accessible to an audience that has not encountered these techniques before.
CommentsTypographic errors corrected. To appear in Analysis Mathematica
