L-functions and prime numbers

数论研究的动机是关于自然数结构的深层问题。 这些问题已经吸引了数学家几千年，并在许多不同的领域中找到了应用，例如，密码学、信号处理和数据分析、通信。这里的一些最基本的研究对象是素数。对素数的研究使用筛子理论的工具，以及与L函数零点的联系。该奖项将使PI能够进一步探索有关L函数和素数家族的基本问题。此外，这将支持PI，因为他继续指导研究生和本科生，组织会议和研讨会，并参加课外活动。 PI将研究L函数的值和零点的分布。我们研究统计二次字符的基本判别式，突出的问题有密切的联系，算术问题。我们扩展Conrey，Iwaniec和Soundararajan的结果的临界线定理的Dirichlet L-功能的建设，我们以前的联合工作的时刻。我们研究自守L-函数的八阶矩，它已经表现出新的功能。我们调查的统计数据的一个大的正交家庭的第一次，在那里我们有一个计划，以“减少导体”，这也导致同时非零的结果。我们攻击的次凸性结果的某些L-功能的导体下降的家庭，这是建立在我们以前的工作时刻。此外，我们还研究了与素数理论的基础知识有关的特殊形式的素数计数问题。该项目由代数与数论计划和刺激竞争研究的既定计划（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智力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.