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。解析数论围绕着一个单一的中心主题，即使用分析技术（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）。我的研究涉及这一中心主题的三个方面：组合数论，乘法组模算术，和零的L-函数。 组合数论关注的是理解整数集合中的加法配置。例如，大的整数集合是否自动包含一个长的算术级数-一个整数序列，其中连续整数之间的间隔都是相同的？如果我们把一个集合中的所有整数对加在一起，我们得到一个新的集合;这个新的集合相对于旧的集合必须有多大？我的研究在一个稍微不同的设置中研究了第二个问题，即整数的有序对，其中只有除以某个素数后的整数被保留。 模算术的核心思想是在被某个固定整数除后只保留余数。如果我们忽略与这个固定整数有共同因子的数字，那么其他的整数可以很好地相乘和相除，形成固定整数的“乘法群”。这种类型的群可以用某些固有量来描述，我的研究给出了关于这些量在乘法群中如何变化的统计信息。 世纪的一个伟大发现是，素数的分布与某些复数函数（称为L函数）取0值的位置密切相关。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。

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.