Moments of Large Families of L-Functions and Related Questions

数学中最著名的猜想之一就是黎曼ζ函数的零点。这种兴趣延伸到l函数的研究，它与许多不同的数学领域，如谐波分析，随机矩阵理论和概率论有联系。在这个领域中，有一个基本的启发式，即l函数的值和零的分布应该匹配随机矩阵的经典紧群的类似统计量。它引发了一系列深层次的猜想，这些猜想在很多层面上仍未得到解决。事实上，即使是这些猜想的特殊情况，也几乎不存在严格的证据。提出的研究将通过证明某些l函数大族的特殊情况来更好地理解这些猜想，这些l函数大族以前在这种情况下没有被理解。该奖项将为研究生和博士后提供研究培训和合作的机会。PI还将利用这笔赠款组织数论研讨会，并指导来自代表性不足群体的学生。这个奖项的目的是研究涉及l函数族的值和零的统计。准确地说，PI的目标是证明l函数大族的高矩的矩猜想的新实例，以及扩展它们的零分布的现有知识（例如n级密度）。此外，还将注意到一些理解高矩的有吸引力的应用，特别是次凸边界和同时不消失的结果。所采用的技术包括传统的傅立叶分析、谱理论和组合数论。该项目由数学科学部的代数和数论项目和促进竞争研究的既定项目（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

One of the most famous conjectures in mathematics is about zeros of the Riemann zeta function. This interest extends to the study of L-functions, which have connections with many diverse areas of mathematics such as harmonic analysis, random matrix theory and probability. In this area, there is a foundational heuristic that the distribution of values and zeros of L-functions should match analogous statistics from classical compact groups of random matrices. It has led to a deep set of conjectures, which remain unresolved at many levels. Indeed, rigorous proofs of even special cases of these conjectures are almost non-existent. The proposed research will lead to a better understanding of these conjectures by proving special cases for certain large families of L-functions that have previously not been understood in this context. The award will provide opportunities for research training and collaboration for graduate students and postdocs. The PI will also use the grant to organize number theory seminars and mentor students from underrepresented groups. The aim of this award is to study statistics involving values and zeros of families of L-functions. To be precise, the PI will aim to prove new instances of the moment conjectures for high moments of large families of L-functions, as well as extend the current knowledge on the distribution of their zeros (e.g. n-level density). Moreover, attention will be paid to certain attractive applications of understanding of high moments, especially subconvexity bounds and simultaneous non-vanishing results. The techniques employed include traditional Fourier analysis, spectral theory, and combinatorial number theory.This project is jointly funded by the Algebra and Number Theory program in the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.