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发挥着核心作用。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。确定素数性质的问题由来已久，从古老的欧几里得定理（素数有无限多个），到 19 世纪中叶黎曼关于 zeta 函数的著名八页论文。从那时起，解析数论中的几个重要问题已经得到解决，而黎曼的思想是这些进步背后的灵感来源。研究黎曼 zeta 函数和 L 函数在不同背景下的性质会导致许多其他有趣的问题，这些问题现在代表了现代数学的重大挑战。事实上，黎曼猜想（断言黎曼 zeta 函数的所有非平凡零点位于特定直线上）和伯奇和斯温纳顿-戴尔猜想（涉及与椭圆曲线相关的 L 函数的某些性质）都已包含在七个千年奖问题中。该项目的目的是研究与黎曼 zeta 函数矩相关的各种问题。这些问题适用于黎曼 zeta 函数零点的间隙分布。该主题与随机矩阵理论（用于描述复杂量子系统的数学物理领域）之间存在着显着的联系。

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.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.