Distribution of Prime Numbers and Related Topics

这个项目的主要目标是提炼Goldston、Pintz和Yildirim最近在几个具体问题上的方法。长期目标是调查一些尚未取得进展的较难解决的问题。Goldston，Pintz和Yildirim的方法通过检查这些序列，当这些序列被某些筛函数加权时，这些筛函数很大，当存在特定构型的素数时，这些筛函数可以通过Selberg筛子或通过优化过程获得。要考察的一些问题是找到可以无条件地通过该方法获得的最小素数间隔大小，确定筛权重如何取决于一个数的素数因数的数量，以及检查某些素数序列以获得正比例结果。这位研究人员还将继续与A.Ledoan一起研究跳跃冠军，研究加性素数理论中的错误项，并研究Zeta函数零点的更高相关性在获得关于重数和零之间差距的信息方面的应用。这个项目与素数有关，数学家们研究了数千年的素数，近年来在计算机科学和密码学中发现了重要的应用。尽管素数的产生过程很简单，但它们提供了一些数学上最具挑战性和最困难的问题，包括黎曼假设(相当于素数具有非常规则的分布)以及Goldbach和Twin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。更广泛地说，数论在整个数学和利用数学的领域都有应用。

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.