Class Groups of Number Fields and Zeros of L-functions

数论是数学的一个分支，研究整数，更具体地说，研究素数。素数定理(描述素数在正整数中的分布)是由Hadamard和de la Vallee Poussin在1896年通过理解Riemann Zeta函数的某些性质而独立证明的。黎曼Zeta函数及其推广，称为L函数，是数论中普遍而又神秘的函数。这些函数可以与过多的数学对象相关联地定义，包括Dirichlet字符、数字字段和椭圆曲线。了解L函数零点的位置是所有数学中的一个中心问题。虽然我们目前还不能证明1859年由黎曼提出的黎曼假设，但为了更好地理解L函数的零点，有许多卓有成效的研究要进行。特别是，L函数零点的垂直分布与另外两个中心问题有很深的联系：类数问题，它始于高斯的工作，以及(广义)黎曼假设的特殊类型反例的可能性。这些假想的反例被称为朗道-西格尔零点，目前还不能排除它们的存在。更具体地说，这个项目将探讨解析数论和代数数论的交叉问题。它将研究关于L函数族的切博塔雷夫密度定理，L函数的零点的垂直分布，以及数域的类数的应用。该项目的活动还将在指导文科背景下的研究生和本科生方面产生更广泛的影响。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Number theory is the branch of mathematics concerned with studying the integers, and more specifically, the primes. The Prime Number Theorem (which describes the distribution of the primes among the positive integers) was proved in 1896 independently by Hadamard and de la Vallee Poussin by understanding certain properties of the Riemann zeta-function. The Riemann zeta-function and its generalizations, called L-functions, are ubiquitous yet mysterious functions in number theory. These functions can be defined in association with a plethora of mathematical objects, including Dirichlet characters, number fields, and elliptic curves. Understanding the location of the zeros of L-functions is a central problem in all of mathematics. While we cannot presently prove the Riemann Hypothesis, posed by Riemann in 1859, there are many fruitful investigations to pursue to better understand the zeros of L-functions. In particular, the vertical distribution of the zeros of L-functions has deep connections to two other central problems: the class number problem, which has its beginnings in the work of Gauss, and the possibility of a special type of counterexample to the (Generalized) Riemann Hypothesis. These hypothetical counterexamples are called Landau-Siegel zeros, and presently their existence cannot be ruled out.More specifically, this project will pursue problems in the intersection of analytic and algebraic number theory. It will study applications related to the Chebotarev density theorem for families of L-functions, the vertical distribution of zeros of L-functions, and class numbers of number fields. The activities of the project will also have broader impacts in terms of mentoring both graduate and undergraduate students in a liberal arts setting.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.