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

Zeta函数几个世纪以来一直在数学中扮演着中心角色，其中最著名的例子是Riemann Zeta函数，它在解析数论和著名的素数定理的证明中起着关键作用，也是著名的Riemann假设的主题。在数论和相关领域中对Zeta函数的研究很自然地导致了Selberg Zeta函数在1956年的引入。具体地说，这是一个单一复变量的函数，类似于Riemann Zeta函数，但它是根据Riemann曲面的几何特征而不是素数来定义的。Selberg Zeta函数相对于Riemann Zeta函数的主要优点之一是对其零点的更明确的刻画。有一种使用Laplace-Beltrami算子的Selberg Zeta函数的经典方法，Laplace-Beltrami算子是一种二阶微分算子，其特征值描述了被视为复函数的Selberg Zeta函数的零点。特别是，在这一背景下，希尔伯特-波利亚算符理论方法对黎曼假设的类比取得了成功。另一方面，在研究Selberg Zeta函数的某些方面，谱方法的成功可以通过不同的方法来扩大。更准确地说，有一种更现代的动力学方法起源于Ruelle的开创性工作，它结合了数学物理、算子理论和遍历理论的成分。在过去的十年里，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用更经典的方法似乎无法获得的结果。例如，这项提议的一个特别主题是周期函数的定义和研究，它与拉普拉斯-贝尔特拉米算符的谱密切相关。虽然使用现有技术很难处理这些重要功能，但本提案的目的是通过沿线开发新的方法来分析这些功能。我们完全希望这项工作将对这一重要数学领域的发展作出重大贡献。

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.