L-functions and prime numbers

数论研究的动机是关于自然数结构的深刻问题。几千年来，这些问题一直吸引着数学家，并在许多不同的领域得到了应用，例如密码学、信号处理和数据分析、通信。这里一些最基本的研究对象是质数。质数的研究使用筛理论的工具，以及与l函数零点的联系。这个奖项将使PI进一步探索关于l函数族和素数的基本问题。此外，这将支持PI继续指导研究生和本科生，组织会议和研讨会，并参与课外活动。PI将研究l函数的值和零的分布。我们研究具有基本判别式的二次元统计，其中突出的问题与算术问题有着密切的联系。我们在前人关于矩的联合工作的基础上，扩展了Conrey， Iwaniec和Soundararajan关于Dirichlet l -函数的临界线定理的结果。我们研究了自同构l函数的第八矩，它已经显示出新的特征。我们首次研究了一个大正交族的统计量，其中我们有一个“减少导体”的计划；这也导致了同时不消失的结果。我们在前人关于矩的研究的基础上，研究了导体掉落族中某些l函数的次凸性结果。进一步研究了与素数理论基本猜想有关的一种特殊形式的素数计数问题。本项目由代数与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。

Research in number theory is motivated by deep questions regarding the structure of the natural numbers. Such questions have fascinated mathematicians for millennia and have found applications in a number of diverse areas, e.g., cryptography, signal processing and data analysis, communications. Some of the most fundamental objects of study here are the prime numbers. The study of prime numbers uses tools from sieve theory, as well as connections to the zeros of L-functions. This award will enable the PI to further explore fundamental questions about families of L-functions and prime numbers. Further, this will support the PI as he continues to mentor graduate and undergraduate students, organizes conferences and seminars, and participates in extra-curricular outreach. The PI will study the distribution of values and zeros of L-functions. We study statistics over quadratic characters with fundamental discriminants, where prominent problems have intimate connections to arithmetic problems. We extend results of Conrey, Iwaniec and Soundararajan on the critical line theorem for Dirichlet L-functions by building on our previous joint work on moments. We study the eighth moment of automorphic L-functions, which has already exhibited novel features. We investigate the statistics of a large orthogonal family for the first time, where we have a plan to “reduce the conductor”; this also leads to simultaneous non-vanishing results. We attack subconvexity results for certain L-functions in conductor dropping families, which builds on our previous work on moments. Further, we study problems in counting prime numbers of a special form, which is connected to foundational conjectures in prime number theory.This project is jointly funded by Algebra and Number Theory Program 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.