RUI: Moments of Short Divisor Sums and the Distribution of Primes

RUI：短除数和的矩和素数分布

• 批准号: 0300563
• 负责人: Daniel Goldston
• 金额: $ 25.57万
• 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: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2008-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0300563&HistoricalAwards=false
• 关键词: RUI; Moments; Short; Divisor; Sums

The investigator in joint work with C. Y. Yildirim has over the last three years determined the correlations of a short divisor sum which approximates the prime numbers. The corresponding correlations for primes were conjectured by Hardy and Littlewood in 1923 and are now referred to as the prime r-tuple conjecture. The proof of this conjecture is probably far beyond our current state of knowledge, but the corresponding result for the short divisor sum is now available. Further, using the Bombieri-Vinogradov theorem one can also obtain all the correlations when one of the divisor sums is replaced by the primes. These two types of correlations together with positivity allow us to obtain information about primes in the form of lower bounds on certain sums over primes. The first and main aim of this project is to apply these new correlation results to the problem of finding small gaps between primes. The main question to be answered is whether one can prove that there are infinitely many prime gaps shorter than any small multiple of the average gap size. At present it is only known that there are small gaps of size less than 0.248 times the average spacing. The method we use is based on moments for short divisor sums in short intervals. Preliminary work indicates we should be able to substantially improve on all previous results. There are many possible refinements and a variety of ways to apply the moment results which will be investigated. The second aim of this project is to refine the correlation results so far obtained and extend their range of applicability. A third aim is to seek further applications of the method to primes in arithmetic progressions and zeros of the Riemann zeta-function.This proposal is concerned with proving results on the distribution of primes. The primes have been a topic of interest since the Greeks who first proved both the infinitude of primes and the unique factorization of all integers into primes. The primes are intimately connected with the Riemann hypothesis and are fundamental objects in both number theory and mathematics. New results and methods concerning primes have been used in many areas of mathematics, physics, and computer science. To cite a recent example, the celebrated proof that primality testing can be done in polynomial time depended on the existence of primes p with p - 1 having a large prime factor, a result proved by Fouvry 17 years ago which up to this year had only an esoteric application to the first case of Fermat's last theorem.

研究者与C. Y. Yildirim已在过去三年确定的相关性短除数和近似的素数。相应的素数相关性由哈代和利特尔伍德在1923年提出，现在被称为素数r元组猜想。这个猜想的证明可能远远超出了我们目前的知识水平，但现在可以得到短因子和的相应结果。此外，使用Bennieri-Vinogradov定理，当其中一个除数和被素数替换时，也可以获得所有相关性。这两种类型的相关性与正性一起，使我们能够以某些素数和的下界的形式获得有关素数的信息。这个项目的第一个和主要目的是将这些新的相关结果应用于寻找素数之间的小间隙的问题。需要回答的主要问题是是否可以证明存在无限多个比平均间隙大小的任何小倍数更短的素数间隙。目前只知道有小于平均间距0.248倍的小间隙。我们使用的方法是基于短间隔短因子和的矩。初步工作表明，我们应该能够大大改善所有以前的结果。有许多可能的改进和各种方法来应用的时刻的结果，将进行调查。本项目的第二个目标是改进迄今为止获得的相关结果，并扩大其适用范围。第三个目的是寻求进一步的应用程序的方法素数的算术级数和零点的Riemann zeta函数。这个建议是有关证明结果的分布素数。素数一直是一个有趣的话题，因为希腊人首先证明了素数的无限性和所有整数到素数的唯一分解。素数与黎曼假设密切相关，是数论和数学的基本对象。有关素数的新结果和新方法已在数学、物理和计算机科学的许多领域得到应用。举一个最近的例子，著名的证明，素性测试可以在多项式时间内完成取决于素数的存在 p 与p - 1有一个大的素数因子，一个结果证明了Fouvry 17年前，直到今年只有一个深奥的应用第一种情况下的费马最后定理。