Analytic Number Theory and mean values of L-functions

黎曼ζ函数和其他l函数在解析数论和一般数学中起着中心作用。例如，黎曼ζ函数满足欧拉积，它强调了自然数和素数之间的联系。确定素数性质的问题有着悠久的历史，从欧几里得关于素数有无穷多个的古老定理，到19世纪中期黎曼关于ζ函数的著名的八页论文。从那时起，解析数论中的几个重要问题得到了解决，黎曼的思想一直是这些进步背后的灵感来源。该项目的目的是研究与黎曼ζ函数和l函数的矩有关的各种问题，这些矩是这些函数的某些族的平均值。这些问题适用于黎曼ζ函数的零点分布（黎曼假设的部分答案），l函数的数量级（林德洛夫假设的部分答案），l函数在中心点的消失顺序（对Birch和Swinnerton-Dyer猜想的分析进展），以及许多其他问题。这门学科与随机矩阵理论（一个用于描述复杂量子系统的数学物理领域）之间有着显著的联系。

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.