RUI: Low-Lying Zeros of L-functions and Problems in Additive Number Theory

RUI：L 函数的低位零点和加法数论中的问题

• 批准号: 1265673
• 负责人: Steven Miller
• 金额: $ 13.56万
• 依托单位: Williams College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265673&HistoricalAwards=false
• 关键词: RUI; Low; Lying; Zeros; functions

The PI plans on studying several projects on zeros of L-functions, as well as problems in additive number theory. The main research concerns the behavior of zeros near the central point. While these zeros have been known to be related to arithmetic problems since Riemann, in the last few decades connections have been observed with high energy nuclear physics and random matrix theory (RMT) as well. Thus investigations in one of these topics can be fruitfully used in the others. The additive number theory problems have a similar flavor; many of them concern density of states as well as gaps between events. Similar techniques are used in the analysis here as in some of the number theory and random matrix theory investigations. Specifically, the PI proposes to study: (1) the n-level densities of low-lying zeros of GL(2) L-functions (including level 1 Maass forms and increasing the support for holomorphic cuspidal newforms by deriving alternatives to the Katz-Sarnak determinantal expansions that are more amenable for comparing number theory and random matrix theory), (2) the n-level densities of Dirichlet and number field L-functions (which involves exploring and extending our understanding of the finer properties of the distribution of certain classes of primes), (3) finding random matrix theory analogues to Rankin-Selberg convolutions (to extend the predictive ability of RMT), (4) additional consequences of the L-function Ratios Conjecture and other arithmetic conjectures, (5) the density of states and behavior of the eigenvalues of structured random matrix ensembles (with special emphasis on the resulting combinatorics, which is frequently related to other problems of interest), (6) generalized Zeckendorf decompositions and the gaps between summands, (7) generalized sum and difference sets (especially phase transitions from different models of randomness, behavior in subsets of highly structured sets, and results in non-abelian cases), and finally (8) Benford's law of digit bias.The central questions in this proposal involve studying how events are distributed in diverse systems, such as energy levels of heavy nuclei, prime numbers and zeros of L-functions, leading digits in sets of data, and summands in generalized Zeckendorf decompositions. Similar to the Central Limit Theorem, there seem to be a few 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 projects have real world applications; for example, the IRS uses Benford?s law to locate corporate tax fraud. Many of these projects have components that are amenable to numerical experimentation; these and tractable special cases will be investigated with undergraduate research assistants. The PI will also continue his extensive work in math education. In addition to providing numerous mentoring opportunities to his students (such as arranging for them to referee for journals, write reviews for MathSciNet, write expository articles for journals, and co-organize AMS special sessions), the PI will also involve them with expanding his math riddles page (http://mathriddles.williams.edu/). This site is frequently one of the top hits when searching for math riddles, and is used in junior high and high schools around the world to excite students to mathematics.

PI 计划研究几个有关 L 函数零点的项目以及加性数论中的问题。主要研究涉及中心点附近零点的行为。虽然自黎曼以来，这些零就被认为与算术问题有关，但在过去的几十年里，人们也在高能核物理和随机矩阵理论 (RMT) 中观察到了它们之间的联系。因此，对这些主题之一的研究可以有效地应用于其他主题。加法数论问题也有类似的特点。其中许多涉及状态密度以及事件之间的差距。这里的分析中使用了与一些数论和随机矩阵理论研究中类似的技术。具体来说，PI 建议研究：（1） GL(2) L 函数的低零点的 n 级密度（包括 1 级 Maass 形式，并通过推导 Katz- 的替代方案来增加对全纯尖头新形式的支持） Sarnak 行列式展开式更适合比较数论和随机矩阵理论），(2) Dirichlet 和数域的 n 级密度L 函数（涉及探索和扩展我们对某些素数类别分布的更精细属性的理解），(3) 寻找与 Rankin-Selberg 卷积类似的随机矩阵理论（以扩展 RMT 的预测能力），(4 ）L 函数比率猜想和其他算术猜想的附加结果，（5）结构化随机矩阵系综的特征值和状态密度（特别强调由此产生的组合，其中经常与其他感兴趣的问题相关），（6）广义泽肯多夫分解和被加数之间的间隙，（7）广义和集与差集（特别是来自不同随机性模型的相变，高度结构化集合子集中的行为以及结果）在非阿贝尔情况下），最后是（8）本福德数字偏差定律。该提案的核心问题涉及研究事件如何在不同系统中分布，例如重核的能级、素数和零L 函数、数据集中的前导数字以及广义 Zeckendorf 分解中的被加数。与中心极限定理类似，似乎存在一些普遍的间距定律来控制这些现象和其他现象；因此，对这些主题之一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分析、傅立叶分析、数论和概率方面的工具和技术。有些项目具有现实世界的应用；例如，美国国税局使用本福德定律来定位公司税务欺诈。其中许多项目都有适合进行数值实验的组件；这些和易于处理的特殊情况将由本科生研究助理进行调查。 PI 还将继续他在数学教育方面的广泛工作。除了为他的学生提供大量指导机会（例如安排他们为期刊审稿、为 MathSciNet 撰写评论、为期刊撰写说明性文章以及共同组织 AMS 特别会议）外，PI 还将让他们参与扩大他的学术生涯。数学谜语页面（http://mathriddles.williams.edu/）。该网站经常是搜索数学谜语时的热门网站之一，并被世界各地的初中和高中用来激发学生的数学兴趣。

项目成果

On Smoothing Singularities of elliptic orbital integrals on GL(n) and Beyond Endoscopy

GL(n) 和超越内窥镜上椭圆轨道积分的平滑奇异性

• 发表时间: 2018-05
• 期刊: Journal of number theory
• 影响因子: 0.7
• 作者: Gonzales, Oscar;Kwan, Chung H;Miller, Steven J;Van Peski, Roger;Wong, Tian A
• 通讯作者: Wong, Tian A

One-level density for holomorphic cusp forms of arbitrary level

任意级全纯尖点形式的一级密度

• DOI: 10.1007/s40993-017-0091-9
• 发表时间: 2016-04-12
• 期刊: Research in Number Theory
• 影响因子: 0.8
• 作者: Owen Barrett;Paul Burkhardt;Jonathan DeWitt;R. Dorward;Steven J. Miller
• 通讯作者: Steven J. Miller

Dimensional Lower Bounds for Falconer type incidence and point configuration theorems

Falconer 型关联定理和点配置定理的维数下界

• 发表时间: 2018-05
• 期刊: Journal d'Analyse Mathematique
• 作者: DeWitt, Jonathan;Ford, Kevin;Goldstein, Eli;Miller, Steven J;Moreland, Gwyneth;Palsson, Eyvindur A;Senger, Steven
• 通讯作者: Senger, Steven

Benford’s law and continuous dependent random variables

本福德定律和连续因随机变量

• DOI: 10.1016/j.aop.2017.11.013
• 发表时间: 2013-09-22
• 期刊: Annals of Physics
• 影响因子: 3
• 作者: Thealexa Becker;David Burt;Taylor C. Corcoran;Ale;er Greaves;er;Joseph R. Iafrate;Joy Jing;Steven J. Miller;Jaclyn D. Porfilio;Ryan Ronan;J. Samranvedhya;F. Strauch;Blaine Talbut
• 通讯作者: Blaine Talbut

Optimal Point Sets determining few distinct triangles,

确定几个不同三角形的最佳点集，

• 发表时间: 2018-05
• 期刊: Integers
• 作者: Epstein, Alyssa;Lott, Adam;Miller, Steven J;Palsson, Eyvindur
• 通讯作者: Palsson, Eyvindur