Analytic Number Theory and mean values of L-functions

黎曼Zeta函数和其他L函数在解析数论和一般数学中起着核心作用。例如，Riemann Zeta函数满足欧拉乘积，该乘积强调了自然数和素数之间的联系。确定素数的性质的问题由来已久，从古老的欧几里得关于有无限多个素数的定理，到19世纪中期Riemann关于zeta函数的著名的8页论文。从那时起，解析数论中的几个重要问题已经得到解决，而黎曼的思想一直是这些进步背后的灵感来源。这个项目的目的是研究与Riemann Zeta-函数和L-函数的矩有关的各种问题，这些函数是这些函数族的平均值。这些问题在黎曼Zeta函数的零点分布(部分回答黎曼假设)、L函数的量级(部分回答林德洛夫假设)、L函数在中心点的消失顺序(对Birch和Swinnerton-Dyer猜想的分析进展)等许多方面都有应用。这门学科和随机矩阵理论之间有一个显著的联系，随机矩阵理论是用于描述复杂量子系统的数学物理领域。

The Riemann zeta-function and other L-functions play a central role in analytic number theory and in mathematics in general. For example, the Riemann zeta-function satisfies an Euler product, which underlines a connection between the natural numbers and the prime numbers. The problem of determining the properties of prime numbers has a long history, from the ancient theorem of Euclid that there are infinitely many primes, to the celebrated eight page paper of Riemann on the zeta-function in the mid-nineteenth century. Since that time, several important problems in analytic number theory have been solved, and Riemann's ideas have been the inspiration behind much of this progress. The aim of the project is to study various questions related to the moments of the Riemann zeta-function and L-functions, which are the mean values over certain families of these functions. These questions have applications to the distribution of zeros of the Riemann zeta-function (partial answers to the Riemann Hypothesis), the order of magnitude of L-functions (partial answers to the Lindelof Hypothesis), order of vanishing of L-functions at the central point (analytic progress towards the Birch and Swinnerton-Dyer Conjecture), and many others. There is a remarkable connection between the subject and Random Matrix Theory, an area of Mathematical Physics used to describe complex quantum systems.