Critical Zeros of L-Functions

这个项目涉及数论的研究，特别是与素数分布有关的问题。由于Zhang和Maynard在连续素数之间的小间隙上的突破性工作，这一主题在过去一年中取得了相当大的进展，其中Fouvry， Iwaniec和Friedlander先前的工作发挥了关键作用。现代数学中最基本的开放问题之一是黎曼假设，它将素数的分布与生成ζ函数的复零分布联系起来。该假设断言所有这些零都位于一条“临界”线上。这个项目的主要目标是表明这些零的很大一部分确实存在于那里。这个结果将增强我们对算术基本元素，素数的微妙行为的理解。PI撰写了与数论相关的几个主题的说明性书籍，最近与约翰弗里德兰德（John Friedlander）合著了一本关于筛法的书。PI积极培养研究生，并计划在该奖项下继续培养研究生。该项目从Levinson-Conrey的方法开始，该方法设计用于估计临界线上黎曼ζ函数的零点百分比。由Conrey建立的zeta函数及其奇阶导数的线性组合具有很强的鲁棒性。当然，这需要一个合适的狄利克雷多项式来缓和。一个人能处理的缓和剂越长，暴露的关键零就越多。该方法的关键创新之处在于将卷积系数特殊安排为“局部缓和”，允许应用很长的全局缓和，从而提高了对现有结果的实质性改进的希望。

This project concerns research in the Theory of Numbers, in particular questions related to the distribution of prime numbers. This topic has advanced considerably during the past year due to the breakthrough work of Zhang and Maynard on small gaps between consecutive prime numbers, where previous work of Fouvry, Iwaniec, and Friedlander played a crucial role. One of the most fundamental open questions in modern mathematics is the Riemann hypothesis, which connects the distribution of prime numbers with the distribution of complex zeros of the generating zeta function. The hypothesis asserts that all these zeros lie on a single "critical" line. The main goal of this project is to show that a large proportion of these zeros are indeed resting there. This result will enhance our understanding of subtle behavior of the basic elements of arithmetic, the prime numbers. The PI has written expository books on several topics related to the Theory of Numbers, most recently a book on Sieve Methods, joint with John Friedlander. The PI is active in training graduate students and plans to continue to train graduate students under this award.The project starts with the method of Levinson-Conrey designed for estimating the percentage of zeros of the Riemann zeta function that are on the critical line. The original construction by Conrey of a linear combination of the zeta function and its odd order derivatives is quite robust. Naturally, this needs to be mollified by a suitable Dirichlet polynomial. The longer mollifier one can handle, the more critical zeros reveal. The key innovation of the proposed approach is a special arrangement of the convolution coefficients into "local mollifiers" which allows applying very long global mollification, thus raising hope for substantial improvements of the existing results.