Analytic Number Theory on Higher Rank Groups

该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，它将素数的分布与生成zeta函数（黎曼zeta函数）的复零点的分布联系起来。黎曼假设断言非整数零位于一条直线上。黎曼Zeta函数是L函数的一个例子。这是一个研究一般L-函数零点的项目。更详细地说，西格尔零点是关于狄利克雷L-函数零点的（广义）黎曼假设的一种潜在反例，与算术级数中素数分布的研究有关。在本计画的主要部分，我们将在Sarnak方法的基础上，使用一种新的技巧来推导一般L-函数的标准无零区域。这种新技术结合了使用的马斯-塞尔伯格关系的高等级群体，预计将删除技术筛参数Sarnak的工作。新的技术也将阐明经典的西格尔零问题，如上所述。其他项目包括研究等分布的马斯形式对更高的秩组和频谱身份的L-功能。这将在解析数论、表示论和遍历理论之间架起新的桥梁。

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.