Gaps Between Primes

素数之间的差距

• 批准号: 0804181
• 负责人: Daniel Goldston
• 金额: $ 21万
• 依托单位: San Jose State University Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2011-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0804181&HistoricalAwards=false
• 关键词: Gaps; Between; Primes

In 2005 the investigator, jointly with Janos Pintz and Cem Yildirim introduced a method that proved for the first time that there exist infinitely often pairs of prime numbers whose difference is smaller than any fraction of the average spacing between primes. The method is especially interesting because it demonstrates that the distribution of primes in arithmetic progressions contains more information on the behavior of primes than had previously been recognized. In particular, if the primes are well distributed in "thin" arithmetic progressions then one can prove that there are always pairs of primes with difference 16 or less. This conditional result shows that the twin prime conjecture - that there are infinitely many primes differing by 2, is approachable with this type of information. More recently the PI in joint work with Yildirim and Pintz and Sid Graham have been working on applying our method to numbers with a specific number of prime factors, especially numbers which have exactly two prime factors. The main goal of this project is to further develop our method to extract the best possible quantitative information on small gaps between primes, and in addition to investigate applications to other problems in number theory. The investigator is also interested in questions related to the use of explicit formulas to study zeros of the Riemann zeta-function.This project is concerned with prime numbers, an ancient subject extending back to the Greeks and up to the present with many 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. Many if not most mathematicians judge the most famous and important unsolved problem in mathematics to be the Riemann Hypothesis, and this is equivalent to the primes being distributed in a fairly regular distribution. More generally, the field of number theory has applications throughout mathematics and fields that make use of mathematics.

2005年，这位研究者与Janos Pintz和Cem Yildirim共同引入了一种方法，首次证明存在无穷多对素数，它们的差小于素数之间平均间隔的任何分数。该方法特别有趣，因为它证明了等差数列中素数的分布包含了比以前认识到的更多关于素数行为的信息。特别是，如果素数在“细”等差数列中分布良好，那么人们就可以证明总有一对素数的差值小于或等于16。这个条件结果表明，双素数猜想——有无穷多个素数差为2——可以用这种类型的信息逼近。最近，PI与Yildirim， Pintz和Sid Graham合作，一直致力于将我们的方法应用于具有特定数量质因数的数字，特别是具有恰好两个质因数的数字。该项目的主要目标是进一步发展我们的方法，以提取质数之间小间隙的最佳定量信息，并研究数论中其他问题的应用。研究者还对使用显式公式来研究黎曼ζ函数的零点的相关问题感兴趣。这个项目与素数有关，这是一个古老的主题，可以追溯到希腊，直到现在，在计算机科学和密码学中有许多重要的应用。尽管素数的产生过程很简单，但它却提出了一些最具挑战性和最困难的数学问题。许多（如果不是大多数的话）数学家认为数学中最著名和最重要的未解决问题是黎曼假设，这相当于质数以相当规则的分布分布。更一般地说，数论领域在整个数学和使用数学的领域都有应用。