RUI: Additive Number Theory, Zeros of L-Functions, and Benford's Law

这个项目的中心问题涉及事件如何在不同的系统中分布，如重核的能级、数据集中的前导数字以及整数中的质数。与概率和统计学中的中心极限定理类似，似乎存在支配这些现象和其他现象的普遍间隔定律；因此，对其中一个主题的研究经常可以为其他主题提供有用的见解。理解这些系统需要开发复数分析、傅立叶分析、数论和概率论方面的工具和技术。其中一些主题具有直接的实际应用；例如，美国国税局使用本福德定律来定位企业税务欺诈。这个项目中研究的许多问题都有适合于数值实验的组成部分；这些和容易处理的特殊情况将由本科生、研究生和博士后研究助理进行调查。调查员还将继续数学教育方面的工作。除了为学生提供职业发展机会(如安排他们担任期刊的评委，为数学评论撰稿，为期刊撰写说明性文章，以及在专业学会会议上共同组织专题会议)，调查员还将让学生参与扩展数学谜语网站(Mathriddles.Williams.edu)，这个网站在世界各地的初中和高中使用，以激发学生对数学的兴趣。这个研究项目研究了L函数、加法数论和本福德定律的各种问题。一个中心主题是分析事件之间的差距。主要内容涉及L函数的零点；人们观察到这些零点与高能核物理和随机矩阵理论之间的联系。正在研究的问题包括：L函数零点的N级密度(主项和低阶项)，更易于在数论和RMT之间进行比较的Katz-Sarnak行列式展开式的替代方案，确定约束超额秩值的最佳测试函数，L函数的傅立叶系数二阶矩的偏差，通过删节的RMT系综对中心点附近的零点进行建模，L函数的零点之间的巨大差距，结构化随机矩阵集合的状态密度和本征值的行为，广义Zeckendorf分解以及和集、广义和集与差集之间的差距，关于有限域和非对易环境中避免三项几何级数的集合的Ramsey理论，以及碎片问题和欺诈检测中的Benford定律。

The central questions in this project concern how events are distributed in diverse systems, such as energy levels of heavy nuclei, leading digits in sets of data, and the prime numbers among the integers. Similar to the central limit theorem in probability and statistics, there seem to be universal spacing laws that govern these and other phenomena; thus studies in one of these topics can frequently provide useful insights in the others. Understanding these systems requires the development of tools and techniques in complex analysis, Fourier analysis, number theory, and probability. Some of the topics have immediate practical applications; for example, the Internal Revenue Service uses Benford's law to locate corporate tax fraud. Many of questions under study in this project have components that are amenable to numerical experimentation; these and tractable special cases will be investigated with undergraduate, graduate, and postdoctoral research assistants. The investigator will also continue work in mathematics education. In addition to providing professional development opportunities to students (such as arranging for them to referee for journals, contribute to Mathematical Reviews, write expository articles for journals, and co-organize special sessions at professional society meetings), the investigator will involve students in expanding the Math Riddles web page (mathriddles.williams.edu), a site that is used in junior high and high schools around the world to excite students about mathematics.This research project studies a variety of problems on L-functions, additive number theory, and Benford's law. A central theme is an analysis of gaps between events. The main topic concerns zeros of L-functions; connections have been observed between these and high energy nuclear physics and random matrix theory (RMT). Among the questions under study are: n-level densities (main and lower order terms) for zeros of L-functions, alternatives to the Katz-Sarnak determinantal expansions that are more amenable for comparisons between number theory and RMT, determining the optimal test functions to bound excess rank, biases in second moments of Fourier coefficients of L-functions, modeling zeros near the central point through excised RMT ensembles, large gaps between zeros of L-functions, the density of states and behavior of the eigenvalues of structured random matrix ensembles, generalized Zeckendorf decompositions and the gaps between summands, generalized sum and difference sets, Ramsey theory for sets avoiding 3-term geometric progressions in finite fields and non-commutative settings, and Benford's law in fragmentation problems and fraud detection.