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L-函数零点的几个项目，以及加法数论中的问题。主要研究了零在中心点附近的行为。虽然这些零点自黎曼以来就被认为与算术问题有关，但在过去的几十年里，高能核物理和随机矩阵理论（RMT）也观察到了它们的联系。因此，在这些主题之一的调查可以卓有成效地用于其他。加性数论问题也有类似的味道;其中许多涉及态密度以及事件之间的间隙。类似的技术被用于分析在这里作为在一些数论和随机矩阵理论的调查。具体而言，PI建议研究：（1）GL（2）L-函数的低零点的n-能级密度（包括1级Maass形式，并通过导出更适合比较数论和随机矩阵理论的Katz-Sarnak行列式展开式的替代方案来增加对全纯尖点新形式的支持），（2）Dirichlet函数和数域L-函数的n层密度（这涉及探索和扩展我们对某些类素数分布的更精细性质的理解），（3）寻找Rankin-Selberg卷积的随机矩阵理论类似物（扩展RMT的预测能力），（4）L函数比率猜想和其他算术公式的附加结果，（5）结构随机矩阵系综的态密度和本征值的性质（特别强调由此产生的组合学，这经常与其他感兴趣的问题有关），（6）广义Zeckendorf分解和被加数之间的间隙，（7）广义和差集（特别是来自不同随机模型的相变，高度结构化集合的子集中的行为，以及非阿贝尔情况下的结果），（8）Benford数位偏差定律。该建议的中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L-函数的零，在数据集的前导数字，和广义Zeckendorf分解的被加数。与中心极限定理类似，似乎有一些普遍的间距定律支配着这些和其他现象;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。其中一些项目有真实的世界应用;例如，美国国税局使用本福德？的法律来定位公司税务欺诈。这些项目中的许多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研究助理一起调查。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的指导机会（例如安排他们为期刊做裁判，为MathSciNet写评论，为期刊写临时文章，以及共同组织AMS特别会议），PI还将让他们参与扩展他的数学谜语页面（http：//mathriddles.wilders.edu/）。这个网站经常是搜索数学谜语的热门之一，并在世界各地的初中和高中使用，以激发学生对数学的兴趣。

项目成果

Benford’s law and continuous dependent random variables

• DOI: 10.1016/j.aop.2017.11.013
• 发表时间: 2013-09
• 期刊: Annals of Physics
• 影响因子: 3
• 作者: Thealexa Becker;David Burt;Taylor C. Corcoran;Alexander Greaves-Tunnell;Joseph R. Iafrate;Joy Jing;Steven J. Miller;Jaclyn D. Porfilio;Ryan Ronan;J. Samranvedhya;F. Strauch;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
• 期刊: Integers
• 作者: Epstein, Alyssa;Lott, Adam;Miller, Steven J;Palsson, Eyvindur
• 通讯作者: Palsson, Eyvindur

One-level density for holomorphic cusp forms of arbitrary level

• DOI: 10.1007/s40993-017-0091-9
• 发表时间: 2016-04
• 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

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
• 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

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
• 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