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

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）以及其他数学领域中有意想不到的应用。解析数论围绕着使用分析技术（如微积分和复数）来建立关于数字（如素数和特殊序列）的信息的单一中心主题。我的研究涉及这个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。******素数的分布与复数密切相关，而复数的某些函数，称为l函数，取0为值。我们对这些零的位置之间的关系知之甚少；例如，两个0的平均值可以是第三个0吗？我的研究将试图证明，这种巧合从未发生过。我们对这些零理解得越好，我们就越能确定哪一组数字比其他集合包含更多的素数，当我们根据它们除以某个固定整数的余数对数字进行分组时。******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论中都具有中心重要性。令人惊讶的是，许多来自数论的数据集也近似于钟形曲线；例如，随机选择一个大的数字并计算它的质因数，结果是一个近似正态分布。我的研究将极大地扩展由数论产生的统计类型，这些统计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加在一起的结果，每个复数都位于同一个圆上，其位置由我们最终试图研究的一组数字决定。我们越了解这些指数和（它们有多大，是否有任何长期趋势），我们就越能了解构成指数和的一组数字。我的研究旨在加强我们对各类指数和的现有知识，所有这些都是为了推动旧的研究项目向新的方向发展

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.**