Zeros of L-functions, and multiplicative and combinatorial number theory

数论是纯数学中最古老的领域之一，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：使用分析技术(如微积分和复数)建立有关数的信息(如质数和特殊序列)。我的研究涉及到这个中心主题的三个方面：组合数论、模算术中的乘群和L函数的零点。 组合数论涉及到理解整数集合中的加法构型。例如，大型整数集是否会自动包含一个长的算术级数--一个连续整数之间的间隔都相同的整数序列？如果我们把一个集合中的所有整数对加在一起，我们就得到了一个新的集合；这个新的集合相对于旧的集合必须有多大？我的研究在一个略有不同的环境中研究了第二个问题，即有序的整数对，其中只保留除以某个质数后的余数。 模算术的核心思想是在除以某个固定整数后只保留余数。如果我们忽略与固定整数有共同因子的数字，那么其他余数可以很好地相乘和除法，形成固定整数的“乘法组”。这种类型的群可以用某些内在量来描述，我的研究给出了这些量在乘性群中如何变化的统计信息。 19世纪最伟大的发现之一是，素数的分布与复数的某些函数，称为L函数，以0为值的位置密切相关。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。

Number theory is one of the most ancient fields of pure 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: combinatorial number theory, multiplicative groups in modular arithmetic, and zeros of L-functions. Combinatorial number theory is concerned with understanding additive configurations in sets of integers. For example, do large sets of integers automatically contain a long arithmetic progression - a sequence of integers where the gaps between consecutive integers are all the same? If we add all pairs of integers from a set together, we get a new set; how large must this new set be in terms of the old one? My research investigates this second question in a slightly different setting, namely ordered pairs of integers where only the remainders after division by some prime number are retained. The idea of retaining only the remainders after division by some fixed integer is the key idea in modular arithmetic. If we ignore numbers that have a factor in common with that fixed integer, then the other remainders can be multiplied and divided nicely, forming the fixed integer's "multiplicative group". Groups of this type can be described by certain intrinsic quantities, and my research gives statistical information on how these quantities vary among multiplicative groups. One of the great discoveries of the 19th century is that the distribution of prime numbers is intimately connected to the places where certain functions of complex numbers, 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.