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

数论是数学中最古老的领域之一，经常有意想不到的应用到真实的世界（如密码学和声学）以及其他数学领域。解析数论围绕着使用分析技术（如微积分和复数）来建立有关数字的信息（如素数和特殊序列）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。素数的分布与复数有着密切的联系，对于复数，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，两个零的平均值是否可能是第三个零？我的研究将试图证明这类巧合从未发生过。我们对这些零的理解越好，当我们根据它们除以某个固定整数后的余数对这些数进行分组时，我们就越能够确定哪些数集比其他数集包含更多的素数。正态分布（或钟形曲线）是对各种数据集的平滑近似，因此在统计和概率中具有核心重要性。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，随机选取一个大数字并计算其素因子，结果近似于正态分布。我的研究将极大地扩展从数论中产生的统计类型，这些统计类型已被证明是正态分布的近似值。指数和是许多复数相加的结果，每个复数都位于同一个圆上，位置由我们最终试图研究的数字集决定。我们对这些指数和（它们有多大，是否有任何长期趋势）的理解越好，我们就越能说指数和是由一组数字构成的。我的研究旨在加强我们对各种类型的指数和的现有知识，所有这些都是为了将旧的研究项目推向新的方向。

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.**