Moments of Large Families of L-Functions and Related Questions

数学中最著名的命题之一是关于黎曼zeta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，如调和分析，随机矩阵理论和概率。在这一领域，有一个基本的启发式，即L-函数的值和零点的分布应该与随机矩阵的经典紧群的类似统计相匹配。它导致了一系列深刻的问题，这些问题在许多层面上仍然没有得到解决。事实上，严格的证明，甚至特殊情况下，这些命题几乎是不存在的。拟议的研究将导致更好地理解这些programmures通过证明特殊情况下，某些大家庭的L-功能，以前没有被理解在这种情况下。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。该奖项的目的是研究涉及L-函数族的值和零点的统计。确切地说，PI的目标是证明大型L函数族的高矩的矩量的新实例，以及扩展当前关于其零点分布的知识（例如n级密度）。此外，注意力将支付给某些有吸引力的应用程序的理解高的时刻，特别是次凸边界和同时非零的结果。所采用的技术包括传统的傅立叶分析，频谱理论，该项目由数学科学部的代数和数论项目以及刺激竞争性研究的既定项目（EPSCoR）共同资助。该奖项反映了NSF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进行评估，被认为值得支持的搜索.

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.