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

该项目的核心问题涉及事件如何在不同系统中分布，例如重核的能级，数据集中的前导数字以及整数中的素数。类似于概率和统计学中的中心极限定理，似乎存在着支配这些和其他现象的普遍的间隔定律;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。理解这些系统需要复杂分析、傅立叶分析、数论和概率的工具和技术的发展。其中一些主题有直接的实际应用;例如，美国国税局使用本福德定律来定位公司税务欺诈。在这个项目中研究的许多问题都有适合数值实验的组件;这些和易处理的特殊情况将与本科生，研究生和博士后研究助理一起调查。研究人员还将继续从事数学教育工作。除了为学生提供专业发展机会（例如安排他们为期刊做裁判，为数学评论做贡献，为期刊撰写临时文章，并在专业协会会议上共同组织特别会议），调查员还将让学生参与扩展数学谜语网页（mathriddles.williams.edu），一个在世界各地的初中和高中使用的网站，以激发学生对数学的兴趣。本研究项目研究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。主要的主题是关于L函数的零点;这些与高能核物理和随机矩阵理论（RMT）之间的联系已经被观察到。正在研究的问题包括：n能级密度（主项和低阶项），Katz Sarnak行列式展开式的替代方案更适合于数论和RMT之间的比较，确定最佳测试函数以限制多余秩，L函数傅立叶系数二阶矩的偏差，通过切除的RMT系综对中心点附近的零进行建模，L-函数的零点之间的大间隙，结构化随机矩阵系综的状态密度和本征值的行为，广义Zeckendorf分解和被和项之间的间隙，广义和差集，在有限域和非交换设置中避免3项几何级数的Ramsey理论，以及破碎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.