Analytic Number Theory and mean values of L-functions

解析数论和 L 函数的平均值

• 批准号: 2660863
• 金额: --
• 依托单位: University of Manchester
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2660863
• 关键词: Analytic; Number; Theory; mean; values

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 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.

Riemann Zeta功能和其他L功能在分析数理论和数学中起着核心作用。例如，Riemann Zeta功能满足Euler产品，该产品强调了自然数和质数之间的联系。从欧几里得的古老定理，有很多数量的素数到著名的《八页》关于Zeta功能在19世纪中叶的著名的Riemann论文。自那时以来，已经解决了分析数理论中的几个重要问题，而Riemann的想法是这一进步的大部分进展的灵感。对Riemann Zeta功能的特性进行评估，在各种情况下，Liemann Zeta功能的属性导致了许多其他有趣的问题，这现在代表了现代数学中的主要挑战。实际上，Riemann假设都断言Riemann Zeta功能的所有非平凡的零位于特定行上，以及桦木和Swinnerton-Dyer猜想涉及与椭圆形曲线相关的L-enguntions的某些特性，这些特性已包含在七千年奖中的问题。 Zeta功能和L功能，这是这些功能某些家族的平均值。这些问题适用于Riemann Zeta功能的零分布（对Riemann假设的部分答案），L功能的数量级（对Lindelof假设的部分答案），在中央点消失的L功能消失的顺序（分析了Birch和Swinnerton-Dyerton-Dyernerton-Dyerton-DyertoRENTIC），以及许多其他定位），以及许多其他。主题与随机矩阵理论之间存在着显着的联系，这是用于描述复杂量子系统的数学物理学领域。