A transfer operator approach to Maass cusp forms and the Selberg zeta function

Maass 尖点形式和 Selberg zeta 函数的传递算子方法

基本信息

批准号：
EP/K000799/1
负责人：
Mark Pollicott
金额：
$ 34.16万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FK000799%2F1
关键词：
transfer operator approach Maass cusp

项目摘要

Zeta functions have played a central role in mathematics for many centuries and the most famous example, the Riemann zeta function, plays a pivotal role in analytic number theory and the proof of the famous Prime Number Theorem, and is also the subject of the famous Riemann Hypothesis. The study of zeta functions in number theory and related fields lead very naturally to the introduction, in 1956, of the Selberg zeta function. In particular, this is a function of a single complex variable which is analogous to the Riemann zeta function, but is defined in terms of characteristics of the geometry of Riemann surfaces rather than in terms of prime numbers. One of the principle advantages of the Selberg zeta function over the Riemann zeta function is the more explicit characterization of its zeros. There is a classical approach to the Selberg zeta function using the Laplace-Beltrami operator, which is a second order differential operator whose eigenvalues describe the zeros of the Selberg zeta function viewed as a complex function. In particular, this is a setting where the Hilbert-Polya operator theoretic approach to the analogue of the Riemann hypothesis has been successful. On the other hand, there are certain aspects of the study of the Selberg zeta function where the successes of the spectral method can be augmented by a different approach. More precisely, there is a more modern dynamical approach originating in the pioneering work of Ruelle which brings together ingredients from mathematical physics, operator theory and ergodic theory. In the past decade, there has been considerable interest in the interplay between the classical and modern approaches to understanding the Selberg zeta function and its connections with number theory.Of particular interest is the use of the dynamical approach to prove results which appear inaccessible by more classical methods. For example, a particular theme of this proposal is the definition and investigation of period functions, which are intimately related to the spectrum of the Laplace-Beltrami operator. Whereas these important functions are fairly intractable using existing techniques, but the aim of this proposal is to analyse them by developing new approaches along the lines. We have every hope that this work will contribute significantly to the development of this important area of mathematics.

Zeta功能在数学中已经在数学中发挥了核心作用，并且最著名的例子是Riemann Zeta功能，在分析数理论中起着关键作用，并且是著名的质数定理的证明，也是著名的Riemann假设的主题。 Zeta在数字理论和相关领域中的功能的研究非常自然地引起了Selberg Zeta功能的引入。特别是，这是一个单个复杂变量的函数，该变量类似于Riemann Zeta函数，但根据Riemann表面的几何特征而不是质数来定义。 Selberg Zeta函数比Riemann Zeta函数的主要优点之一是其零的更明确表征。使用Laplace-Beltrami操作员，有一种经典的方法，用于Selberg Zeta函数，这是二阶差异操作员的特征值，其特征值描述了Selberg Zeta函数的零，被视为复杂函数。特别是，在这种情况下，希尔伯特 - 帕利亚操作员理论方法对Riemann假设的类似物进行了成功。另一方面，Selberg Zeta函数的研究存在某些方面，其中光谱方法的成功可以通过不同的方法来增强。更确切地说，有一种更现代的动力学方法，起源于Ruelle的开拓性工作，该方法将数学物理学，操作者理论和千古理论的成分汇集在一起。在过去的十年中，人们对了解Selberg Zeta函数及其与数字理论的联系之间的相互作用引起了极大的兴趣。特别是使用动态方法证明结果的结果，这些方法证明了通过更古典的方法无法访问的结果。例如，该提案的一个特定主题是对时期函数的定义和研究，这些定义与Laplace-Beltrami操作员的频谱密切相关。尽管这些重要功能使用现有技术相当棘手，但是该提案的目的是通过沿线开发新方法来分析它们。我们希望这项工作将为数学的这一重要领域的发展做出重大贡献。