Distribution of Prime Numbers and Related Topics

这个项目的主要目标是在几个具体问题中改进Goldston，Pintz和Yildirim最近的方法。长期目标是调查尚未取得进展的一些较困难的问题。Goldston、Pintz和Yildirim的方法通过检查这些序列被某些筛选函数加权时检测到的素数或几乎素数，这些筛选函数在指定配置中存在素数时是大的，这些筛选函数可以通过Selberg筛选或通过优化过程获得。要研究的一些问题是找到最小的素数差距大小，可以无条件地获得的方法，以确定如何筛选权重的行为取决于数量的素因子，并检查某些序列的素数的正比例的结果。调查人员还将继续他的工作，跳跃冠军与A。Ledoan，研究加法素数理论中的误差项，并研究zeta函数的零的更高相关性在获得关于多重性和零之间的间隙的信息中的应用。这个项目关注的是素数，数学家已经研究了数千年，近年来在计算机科学和密码学中发现了重要的应用。尽管它们的产生方式很简单，素数提供了一些最具挑战性和最困难的数学问题，包括黎曼假设（相当于素数具有非常规则的分布）和哥德巴赫和孪生素数猜想。 对这些问题的研究不仅促进了数论的发展，也促进了分析和代数的发展。更一般地说，数论在整个数学和利用数学的领域中都有应用。

The main goal of this project is to refine the recent method of Goldston, Pintz, and Yildirim in several specific problems. The long-term goal is to investigate some of the more difficult questions for which no progress has yet been made. The method of Goldston, Pintz and Yildirim detects primes or almost primes close together by examining these sequences when weighted by certain sieve functions which are large when there are primes in specified configurations.These sieve functions can be obtained by the Selberg sieve or by an optimization process. Some questions to be examined are to find the smallest prime gap size that can be obtained unconditionally by the method, to determine how the sieve weights behave depending on the number of prime factors a number has, and to examine certain sequences of primes for positive proportion results. The investigator will also continue his work on jumping champions with A. Ledoan, work on error terms in additive prime number theory, and examine applications of higher correlation of zeros of zeta functions to obtaining information on multiplicity and gaps between zeros.This project is concerned with prime numbers, which have been studied by mathematicians for thousands of years, and in recent years have found important applications in computer science and cryptography.Despite the simplicity of how they arise, prime numbers offer some of the most challenging and difficult problems in mathematics, including the Riemann Hypothesis (which is equivalent to the primes having a very regular distribution) and the Goldbach and Twin Prime Conjectures. The study of these questions has contributed to the development of not just number theory but analysis and algebra. More generally, number theory has applications throughout mathematics and fields that make use of mathematics.