Analytic Number Theory on Higher Rank Groups

这个项目涉及数论的研究，特别是与素数分布有关的问题。现代数论中最基本的开放问题之一是黎曼假设，它将质数的分布与生成ζ函数的复零分布联系起来，即黎曼ζ函数。黎曼假设断言非整数零位于一条直线上。黎曼函数是l函数的一个例子。这是一个研究一般l函数零点的课题。更详细地说，西格尔零是与研究等差数列中素数分布有关的Dirichlet l -函数的零的（广义）黎曼假设的一种潜在反例。在本项目的主要部分，基于Sarnak的方法，将使用一种新的技术来推导一般l函数的标准无零区域。这项新技术结合了Maass-Selberg关系在高阶群上的应用，有望消除Sarnak工作中的技术筛选论点。新技术也将阐明经典的西格尔零问题，如上所述。其他项目包括研究高阶群上质量形式的等分布和l函数的谱恒等式。这将在解析数论、表示理论和遍历理论之间提供新的桥梁。

This project concerns research in the theory of numbers, in particular questions related to the distribution of prime numbers. One of the most fundamental open questions in modern number theory is the Riemann hypothesis, which connects the distribution of prime numbers with the distribution of complex zeros of the generating zeta function, the Riemann zeta function. The Riemann hypothesis asserts that the non-integer zeros lie on a line. The Riemann Zeta function is an example of an L-function. This is a project to study zeroes of general L-functions. In more detail, a Siegel zero is a type of potential counterexample to the (generalized) Riemann hypothesis on the zeros of Dirichlet L-functions connected to the study of the distribution of primes in arithmetic progressions. In the main part of this project, based on Sarnak's method, a novel technique will be used to derive the standard zero-free regions for general L-functions. This new technique combines the use of Maass-Selberg relations on higher rank groups which is expected to remove the technical sieve arguments in Sarnak's work. The new technique will also shed light on the classical Siegel zero problem as mentioned above. Other projects include the study of equidistribution of Maass forms on higher rank groups and spectral identities for L-functions. These will provide new bridges between analytic number theory, representation theory and ergodic theory.