Frontiers of Number Theory

关于正整数的性质的问题，特别是关于整数的因式和素数的分布的问题，数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得到了应用。这项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差距分布方面取得了重大突破，并将沿着这些路线进行进一步的研究，以及由我们的技术产生的关于素数和组合对象的其他问题。主要研究人员还将继续计算某些类型的方程组、数论函数和Riemann Zeta函数的积分解的工作。这项研究涉及数论中的几个项目。第一个涉及精确计算某些特殊类型的丢番图方程组的整数解，这些方程组是Vinogradov系统的多维类似物。主要目标是为一大类这样的系统获得解的数目的接近最佳的可能上界。第二个主要项目是改进素数之间的大间隙和素数之间的连续间隙链的下界，并获得关于这些间隙中的整数的更多信息。主要研究人员还将研究与大模算术级数中素数的分布和超图覆盖的效率有关的问题，这两个工具都是研究素数间隙的重要工具。进一步的项目包括改进Carmichael函数所取不同值的计数函数，以及研究Riemann Zeta函数的零点的模1分布。

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.