The Distribution of Zeros of L-Functions and Related Questions

L函数在数论的现代发展中起着举足轻重的作用，可以用来研究各种各样的问题。用于研究L函数的工具涉及分析、代数、代数几何、表示论和数学物理等许多数学分支，而数论在数学以外的领域有重要的应用，如理论计算机科学和密码学。本课题研究的许多课题都与L函数的零点分布以及算术中的相关课题有关。这一关系是七个千年奖问题中的两个问题的核心，即黎曼假说和伯奇和斯温纳顿-戴尔猜想。研究人员将继续在这一研究领域对研究生进行培训和指导，该项目将为他们提供研究培训机会。这个项目中的一个问题将调查狄里克莱特·L函数的一个细家族，它在实验上对其零点之间的间隙分布具有显著的、以前未被检测到的偏差。这种偏差似乎有一种算术解释，它对应于与这些L函数的矩有关的某一高斯型和的非零性。其他问题研究黎曼Zeta函数的零点和矩的分布，这是L函数的典型例子，给出了以前没有解释的，在某些情况下以前没有观察到的现象的见解。研究人员还打算将傅立叶分析的工具与L函数理论结合起来研究素数测地线，并在假设广义黎曼假设的情况下，对素数理论中的几个条件估计进行改进。这一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智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

L-functions have played a pivotal role in the modern development of number theory and they can be used to study a wide variety of problems. The tools used to study L-functions draw from many branches of mathematics including analysis, algebra, algebraic geometry, representation theory, and mathematical physics while number theory has important applications outside of mathematics to fields such as theoretical computer science and cryptography. Many of the topics investigated in this project concern the distribution of zeros of L-functions and related topics in arithmetic. This relationship is central to two of the seven Millennium Prize Problems, the Riemann hypothesis and the Birch and Swinnerton-Dyer conjecture. The investigator will continue training and mentoring graduate students in this research area, and this project will provide research training opportunities for them. One of the problems in this project will investigate a thin family of Dirichlet L-functions which experimentally has significant and previously undetected bias in distribution of the gaps between their zeros. This bias seems to have an arithmetic explanation that corresponds to the non-vanishing of a certain Gauss type sum connected to moments of these L-functions. Other problems study the distribution of zeros and moments of the Riemann zeta-function, the prototypical example of an L-function, giving insight on previously unexplained, and in some cases previously unobserved, phenomena. The investigator also intends to combine tools from Fourier analysis with the theory of L-functions to study prime geodesics and to give improvements of several conditional estimates in prime number theory, assuming the generalized Riemann hypothesis.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.