Sieves and primes

几千年来，正整数的性质问题一直吸引着人们，最近在计算机科学、信息安全和信号处理中得到了应用。在这个被称为数论的领域里，素数扮演着中心角色。基本问题围绕着素数的模式、素数之间的间隔以及特殊类型的素数是如何分布的。自20世纪初以来，筛法一直是我们分析这些问题的主要工具之一，但人们对这些方法的局限性知之甚少，发现这些局限性是该领域的一个主要开放问题。这个奖项将使PI继续他的工作，理解和探索筛法和素数的分布。这一努力的成功将有助于解开质数的许多谜团，并对数学和信息论的许多领域产生重大影响。拨款还将用于培训和指导从事数论工作的研究生。PI将开发新的方法来探测筛法在一般整数序列中检测素数的局限性。重点将放在发展一种新的、统一的筛理论上，这种理论允许人们判断序列上的主要假设，即所谓的i型界和ii型界，是否足以证明序列包含许多素数。这些假设的强度由三个参数决定。特别地，我们将首次证明，在这些参数的一定范围内，存在满足Type-I和Type-II界但不包含素数的序列。主要目标是精确地确定在哪个参数范围内，主要假设意味着序列总是包含许多素数。我们还将更详细地研究在短间隔内计数素数的问题。PI还将继续研究进一步理解整数除数的浓度。其主要目的是精确地确定典型整数除数的浓度函数。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Questions about properties of the positive integers have fascinated people for thousands of years and have recently found applications in computer science, information security and signal processing. In this field, known as number theory, the prime numbers play a central role. Fundamental questions revolve around patterns in the primes, gaps between primes, and how primes of special types are distributed. Since the early 20th century, sieve methods have been one of the chief tools we have for analyzing these problems, but the limitations of these methods are poorly understood, and discovering the limitations is a major open problem in the field. This award will enable the PI to continue his work understanding and exploring sieve methods and the distribution of primes. Success in this endeavor will help unlock many of the mysteries of prime numbers and have a significant impact on many areas of mathematics and information theory. Grant funds will also be used to train and mentor graduate students working in number theory. The PI will develop new methods of probing the limitations of sieve methods for detecting primes in a general sequence of integers. The emphasis will be on developing a new, unified theory of sieves that allows one to say if the main hypotheses on the sequence, known as Type-I bounds and Type-II bounds, are sufficient to show that the sequence contains many primes. The strength of these hypotheses are governed by three parameters. In particular, we will prove, for the first time, that in a certain range of these parameters, there are sequences which satisfy the Type-I and Type-II bounds yet contain no primes. The primary goal is to determine precisely in which range of the parameters the main hypotheses imply that the sequence always contains many primes. We will also investigate in finer detail the problem of counting primes in short intervals. The PI will also continue his investigations into further understanding the concentration of divisors of integers. The primary goal is to determine precisely the measure of the concentration function of divisors of typical integers.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.