Frontiers of Number Theory

数论前沿

• 批准号: 1501982
• 负责人: Kevin Ford
• 金额: $ 36万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-05-15 至 2019-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501982&HistoricalAwards=false
• 关键词: Frontiers; Number; Theory

Questions about properties of positive integers, especially the way in which integers factor and the distribution of prime numbers, have fascinated people for thousands of years and have recently found applications in computer science, information security, and signal processing. This proposal concerns several projects in the theory of numbers, emphasizing connections with other areas of mathematics such as Probability and Combinatorics. The principal investigator was part of a team that made a recent large breakthrough on the distribution of gaps between prime numbers, and further investigations will be made along these lines, together with additional problems about primes and combinatorial objects arising from our techniques. The principal investigator will also continue work on counting integral solutions of certain types of systems of equations, number theoretic functions, and the Riemann zeta function.This research concerns several projects in the theory of numbers. The first concerns accurately counting the integer solutions of certain special types of systems of Diophantine equations that are multidimensional analogs of Vinogradov's system. The chief goal is to obtain near best possible upper bounds for the number of solutions of such systems, for a large class of such systems. The second major project is to improve lower bounds for large gaps between prime numbers and chains of consecutive gaps between primes, and gain more information about the integers within such gaps. The principal investigator will also investigate related problems about the distribution of primes in arithmetic progressions to large moduli and the efficiency of hypergraph coverings, both important tools in research on prime gaps. Further projects include refining the counting function of distinct values taken by Carmichael's function, and studying the distribution modulo 1 of the zeros of the Riemann zeta function.

关于正整数的性质的问题，特别是整数因子和素数分布的方式，几千年来一直吸引着人们，最近在计算机科学，信息安全和信号处理中找到了应用。这个建议涉及数论中的几个项目，强调与其他数学领域的联系，如概率和组合学。 首席研究员是一个团队的成员，该团队最近在素数之间的间隙分布方面取得了重大突破，并将沿着这些方向进行进一步的研究，以及由我们的技术产生的有关素数和组合对象的其他问题。主要研究者还将继续研究某些类型的方程组，数论函数和黎曼zeta函数的积分解的计数。这项研究涉及数论中的几个项目。第一个问题是精确计算某些特殊类型的丢番图方程系统的整数解，这些丢番图方程是维诺格拉多夫系统的多维类似物。 主要的目标是获得接近最好的可能的上限的数量的解决方案，这样的系统，为一大类这样的系统。 第二个主要项目是改进素数之间的大间隔和素数之间的连续间隔链的下界，并获得有关此类间隔内整数的更多信息。 首席研究员还将研究有关素数在算术级数中的分布以及超图覆盖的效率的相关问题，这两个问题都是研究素数缺口的重要工具。进一步的项目包括细化计数功能的不同值所采取的卡迈克尔的功能，并研究分布模1的零点的黎曼zeta函数。

项目成果

Extremal Properties of Product Sets

产品集的极值属性

• DOI: 10.1134/s0081543818080175
• 发表时间: 2018
• 期刊: Proceedings of the Steklov Institute of Mathematics
• 影响因子: 0.5
• 作者: Ford, Kevin