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得到了解决，而黎曼的思想一直是这些进展背后的灵感来源。研究黎曼Zeta函数和L函数在不同背景下的性质导致了许多其他有趣的问题，这些问题现在是现代数学中的主要挑战。事实上，黎曼假设，即黎曼Zeta函数的所有非平凡零点都位于一条直线上，以及Birch和Swinnerton-Dyer猜想，涉及与椭圆曲线有关的L函数的一些性质，都被包括在七个千年奖问题中。本项目的目的是研究与黎曼Zeta函数矩有关的各种问题。这些问题在Riemann Zeta函数的零点间隙分布中有应用。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.Investigating the properties of the Riemann zeta-function and L-functions in various contexts leads to many other interesting problems, which now represent major challenges in modern mathematics. In fact both the Riemann Hypothesis, which asserts that all the non-trivial zeros of the Riemann zeta-function lie on a particular line, and the Birch and Swinnerton-Dyer Conjecture, which concerns some properties of the L-functions associated to elliptic curves, have been included in the seven Millennium Prize Problems.The aim of the project is to study various questions related to the moments of the Riemann zeta-function. These questions have applications to the gap distribution of zeros of the Riemann zeta-function. There is a remarkable connection between the subject and Random Matrix Theory, an area of Mathematical Physics used to describe complex quantum systems.