Problems in Analytic Number Theory

解析数论中的问题

• 批准号: 0653529
• 负责人: Hugh Montgomery
• 金额: $ 14.17万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0653529&HistoricalAwards=false
• 关键词: Problems; Analytic; Number; Theory

A wide variety of problems of multiplicative number theory will be pursued.The problem of gaps between sums of two squares will be tackled in a new way,which if successful could be developed into a major new tool for the study ofthe coefficients of automorphic functions in short intervals. Sieve methodswill be studied not by {\it ad hoc} choices of sifting functions but rather byallowing the sieve to reveal where its extremal configurations lie. The goal isto determine, in all dimensions, the optimal upper and lower bounds. The localdistribution of zeros of the zeta function will be studied by locating optimalkernels to use in conjunction with pair correlation information. Statistics relating to the distribution of primes in arithmetic progressions asone averages over different arithmetic progressions will be determined. For several decades, Hardy and Littlewood maintained a list of researchproblems. Among the problems remaining on their final list is toderive a better upper bound for the gap between numbers that can be expressedas a sum of two squares. Here better means simply better than the trivialbound one obtains by the greedy algorithm. This problem is to be attackedin a new way, and if the approach is successful, then the method may applyin greater generality. As was pointed out by Selberg, the problem of sieving efficiently isfundamentally a problem of linear programming. Existing sieves aresomewhat ad hoc, and in most cases the bound obtained is either notoptimal or at least not known to be optimal. It is proposed toemphasize the linear programming aspect of the problem, and to identifyextremals, for both the primal and the dual problems. This is to bedone first in the simplest of situations, and then in successively morechallenging ones, so that eventually one will be able to identifyextremals in realistic problems of great interest.

我们将继续研究乘法数论的各种问题，并将以一种新的方式解决两个平方和之间的差距问题，如果成功，它将发展成为研究短区间内自同构函数系数的一个重要的新工具。筛子方法的研究不是通过选择筛分函数，而是通过允许筛子揭示其极端构型所在。目标是在所有维度中确定最优的上下限。Zeta函数的零点的局部分布将通过定位与配对相关信息一起使用的最优核来研究。关于素数在算术级数中的分布作为不同算术级数的一个平均值的统计将被确定。几十年来，哈代和利特尔伍德一直在维护一份研究问题的清单。在他们的最终清单上剩下的问题之一是推导出数字之间的差距的一个更好的上界，它可以表示为两个平方的和。在这里，更好的意思仅仅是比贪婪算法得到的平凡界限更好。这个问题需要用一种新的方法来处理，如果这种方法是成功的，那么这种方法可能适用于更普遍的情况。正如Selberg所指出的，有效筛选的问题从根本上讲是一个线性规划问题。现有的筛子是临时的，在大多数情况下，得到的界要么不是最优的，要么至少不是已知的最优的。对于原始问题和对偶问题，建议强调问题的线性规划方面，并识别极值。这将首先在最简单的情况下进行，然后在接踵而至的更具挑战性的情况下进行，以便最终能够在人们非常感兴趣的现实问题中识别极端问题。