Frontiers of Number Theory

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得到了应用。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，如概率论和组合论。首席研究员是最近在质数间隙分布方面取得重大突破的团队的一部分，并将沿着这条线进行进一步的研究，以及我们的技术产生的关于质数和组合对象的其他问题。首席研究员还将继续研究某些类型的方程组、数论函数和黎曼ζ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芬图方程系统的整数解，这些系统是维诺格拉多夫系统的多维类似物。主要目标是为这类系统的一大类求得解的数目的接近最佳可能上界。第二个主要项目是改进素数之间大间隙和素数之间连续间隙链的下界，并获得关于这些间隙内整数的更多信息。主要研究者还将研究等差数列中素数的大模分布和超图覆盖的效率等相关问题，这两个问题都是研究素数间隙的重要工具。进一步的项目包括改进Carmichael函数取不同值的计数函数，以及研究Riemann zeta函数零点的分布模1。

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.