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

在过去的三年里，这位研究人员与C.Y.Yildirim共同工作，确定了一个接近质数的短除数和的相关性。素数的相应关联式是Hardy和Littlewood在1923年猜想的，现在被称为素数r-字节组猜想。这一猜想的证明可能远远超出了我们目前的知识状态，但短除数和的相应结果现在已经可用。此外，利用Bombieri-Vinogradov定理，当一个因子和被素数代替时，也可以得到所有的相关性。这两种类型的相关性和正性一起允许我们以某些和对素数的下界的形式获得关于素数的信息。这个项目的第一个也是主要的目的是将这些新的相关结果应用到寻找素数之间的小间隙的问题上。要回答的主要问题是，是否能证明存在比平均间隙大小的任何小倍数都短的无穷多个素数间隙。目前只知道存在大小不到平均间距0.248倍的小间隙。我们使用的方法是基于短区间内的短除数和的矩。初步工作表明，我们应该能够在以前所有结果的基础上有实质性的改进。有许多可能的改进和各种方法来应用将被研究的矩结果。该项目的第二个目标是完善迄今所获得的对比结果，并扩大其适用范围。第三个目标是寻求该方法在算术级数中的素数和Riemann Zeta函数的零点上的进一步应用。这项建议涉及关于素数分布的证明结果。自希腊人首次证明了素数的无穷性和将所有整数分解成素数的唯一性以来，素数一直是人们感兴趣的话题。素数与黎曼假设密切相关，是数论和数学中的基本对象。有关素数的新结果和新方法已被用于数学、物理和计算机科学的许多领域。举一个最近的例子，素性测试可以在多项式时间内完成的著名证明依赖于素数p的存在，其中p-1具有大的素数因子，这一结果在17年前由傅夫里证明，直到今年，它还只深奥地应用于费马大定理的第一种情况。

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.