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 函数在许多世纪以来在数学中发挥着核心作用，最著名的例子是黎曼 Zeta 函数，它在解析数论和著名素数定理的证明中发挥着举足轻重的作用，也是著名的黎曼 Zeta 函数的主题假设。数论和相关领域中 zeta 函数的研究很自然地导致了 1956 年 Selberg zeta 函数的引入。特别是，这是单个复变量的函数，类似于黎曼 zeta 函数，但是根据黎曼曲面的几何特征而不是素数来定义的。塞尔伯格 zeta 函数相对于黎曼 zeta 函数的主要优点之一是其零点的表征更加明确。 Selberg zeta 函数有一种使用 Laplace-Beltrami 算子的经典方法，该算子是二阶微分算子，其特征值描述了被视为复函数的 Selberg zeta 函数的零点。特别是，在这种情况下，希尔伯特-波利亚算子理论方法与黎曼假设的类似物已经取得了成功。另一方面，Selberg zeta 函数研究的某些方面可以通过不同的方法来增强谱法的成功。更准确地说，有一种更现代的动力学方法，它起源于 Ruelle 的开创性工作，它汇集了数学物理、算子理论和遍历理论的成分。在过去的十年中，人们对理解 Selberg zeta 函数及其与数论的联系的经典方法和现代方法之间的相互作用产生了相当大的兴趣。特别感兴趣的是使用动力学方法来证明更多人无法获得的结果。经典方法。例如，该提案的一个特定主题是周期函数的定义和研究，这与拉普拉斯-贝尔特拉米算子的谱密切相关。尽管使用现有技术来处理这些重要的功能相当困难，但本提案的目的是通过开发新的方法来分析它们。我们完全希望这项工作将为数学这一重要领域的发展做出重大贡献。