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，黎曼zeta函数满足欧拉积，它强调了自然数和素数之间的联系。确定素数的性质的问题有着悠久的历史，从欧几里得的古代定理，有无穷多个素数，到著名的八页纸的黎曼在ζ函数在世纪中期。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。在各种背景下研究黎曼zeta函数和L函数的性质导致了许多其他有趣的问题，这些问题现在代表了现代数学的主要挑战。事实上，黎曼猜想（Riemann Hypothesis）和Birch和Swinnerton-Dyer猜想（Birch and Swinnerton-Dyer Conjecture）都包含在七个千禧年奖问题中，其中黎曼猜想断言黎曼zeta函数的所有非平凡零点都位于一条特定的直线上，而Birch和Swinnerton-Dyer猜想则涉及与椭圆曲线相关的L函数的一些性质。这些问题对Riemann zeta函数零点的差距分布有应用。这个主题与随机矩阵理论（Random Matrix Theory）之间有着显着的联系，随机矩阵理论是数学物理学中用于描述复杂量子系统的一个领域。

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.