Dirichlet L-functions, Erdos-Kac theorems, and applications to number theory

Dirichlet L 函数、Erdos-Kac 定理以及在数论中的应用

• 批准号: RGPIN-2016-03756
• 负责人: Martin, Greg
• 金额: $ 1.97万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=614607
• 关键词: Dirichlet; functions; Erdos; Kac; theorems

Number theory is one of the most ancient fields of mathematics, often having unexpected application to the real world (such as cryptography and acoustics) as well as other fields of math. Analytic number theory revolves around the single central theme of using the techniques of analysis (such as calculus and complex numbers) to establish information about numbers (such as prime numbers and special sequences). My research involves three aspects of this central theme: zeros of L-functions, normal distribution theorems, and exponential sums. The distribution of prime numbers is intimately connected to the complex numbers for which certain functions, called L-functions, take 0 as a value. Very little is known about the relationships between the locations of these zeros; for example, can the average of two zeros ever be a third zero? My research will attempt to show that coincidences of this sort never occur. The better we understand these zeros, the more we will be able to determine which sets of numbers contain more prime numbers than other sets, when we group the numbers according to their remainder upon division by some fixed integer. The normal distribution (or bell curve) is a smooth approximation to a wide variety of data sets, and thus has central importance in both statistics and probability. Surprisingly, many data sets coming from number theory are also approximated by bell curves; for example, picking a large number at random and counting its prime factors results in an approximately normal distribution. My research will greatly expand the types of statistics arising from number theory that have been shown to be approximated by normal distributions. An exponential sum is the result of adding together many complex numbers, each of which sits on the same circle, at positions determined by the set of numbers we are ultimately trying to study. The better we understand these exponential sums (how large they are, whether there is any long-term trend), the more we can say about the set of numbers the exponential sum is constructed from. My research aims to strengthen our existing knowledge about various types of exponential sums, all for the purpose of pushing old research projects in novel directions.

数论是最古老的数学领域之一，经常在现实世界(如密码学和声学)以及其他数学领域有意想不到的应用。解析数论围绕一个中心主题，即使用分析技术(如微积分和复数)来建立有关数的信息(如质数和特殊序列)。我的研究涉及到这一中心主题的三个方面：L函数的零点、正态分布定理和指数和。 素数的分布与某些称为L函数的函数以0为值的复数密切相关。人们对这些零的位置之间的关系知之甚少；例如，两个零的平均值会是第三个零吗？我的研究将试图证明，这种巧合永远不会发生。我们对这些零理解得越好，当我们根据数字除以某个固定整数时的余数对数字进行分组时，我们就越能确定哪些数字集比其他集合包含更多的质数。 正态分布(或钟形曲线)是对各种数据集的平滑近似，因此在统计学和概率中都具有核心重要性。令人惊讶的是，许多来自数论的数据集也是由钟形曲线近似的；例如，随机选取一个大的数字并计算其素因数会得到一个近似正态分布。我的研究将极大地扩展数论产生的统计类型，这些类型已经被证明是由正态分布近似的。 指数和是将许多复数相加的结果，每个复数都位于同一个圆圈上，位置由我们最终试图研究的一组数字确定。我们对这些指数和(它们有多大，是否存在任何长期趋势)理解得越好，我们就越能对构成指数和的数字集有更多的了解。我的研究旨在加强我们现有的关于各种类型的指数和的知识，所有这些都是为了推动旧的研究项目朝着新的方向发展。