Analytic Theory of L-functions

l -函数是解析数论中编码算术信息的基本对象。最简单的例子是黎曼函数，狄利克雷l函数，以及与模形式相关的l函数。理解这些l函数的值和零的统计行为是Conrey博士的主要主题。年代的研究。他和他的合作者提出了各种各样的项目，每个项目都涉及l函数的某些方面。其中一个涉及到一个带有除数函数d(n)的新公式，它推广了经典Voronoi公式，并在黎曼假设的Nyman-Beurling方法中得到了应用；Conrey将进一步开发在矩公式和低次l函数理解问题上的应用。Conrey将使用渐近大筛（与Iwaniec和Soundararajan发明）来证明一个关于所有狄利克雷l函数乘以任意狄利克雷多项式的均方的猜想。第三个项目涉及Mazur。关于椭圆曲线的对称幂l函数很少在其中心点消失的猜想。康瑞教授的研究领域是数论。现代数论有着令人惊讶的多种应用，从实现安全的互联网交易，到构建最优网络，甚至到在低维拓扑研究中对各种类型的物体进行分类。数论学家发明的最成功的工具之一是ζ函数。它最初的目的是帮助研究质数。现在，它和它的类似物，在数论中无处不在。然而，我们仍然不了解ζ函数的一些非常基本的性质，如果我们了解了，就会有很大的进展。主要的问题是为什么函数的所有0都出现在一行上？Conrey教授的研究集中在零点函数的研究上。作为该项目的一部分，Conrey教授还将继续他在数学教师圈的工作，这是一个由全国39个问题解决小组组成的集合，其中包括专业数学家和中学数学教师，他们共同建立了问题解决者社区。

L-functions are fundamental objects in analytic number theory which encode arithmetic information. The simplest examples are the Riemann zeta-function, Dirichlet L-functions, and the L-functions associated with modular forms. Understanding the statistical behavior of the values and zeros of these L-functions is the primary theme of Dr. Conrey.s research. He and his collaborators propose a variety of projects each involved with some aspect of L-functions. One involves a new formula with the divisor function d(n) which generalizes the classical Voronoi formula and has an application in the Nyman-Beurling approach to the Riemann Hypothesis; Conrey will develop further applications to moment formulae and to the question of understanding low degree L-functions. Conrey will use the Asymptotic Large Sieve (invented with Iwaniec and Soundararajan) to prove a conjecture about the mean square of all Dirichlet L-functions multiplied by an arbitrary Dirichlet polynomial. A third project involves Mazur.s conjecture that the symmetric power L-functions associated with an elliptic curve seldom vanish at their central point.Professor Conrey's research is in the area of number theory. Modern Number Theory has surprisingly diverse applications, from enabling secure internet transactions, to the construction of optimal networks, and even to the question of cataloguing the various types of bodies in the study of low dimensional topology. One of the most successful tools invented by number theorists is the zeta-function. Its original purpose was to help with the study of prime numbers. Now it, and its analogues, are ubiquitous in number theory. However, there are still some very basic properties of zeta-functions which we do not understand, and which if we did would lead to much progress. The main question is Why do all of the zeros of zeta-functions occur on just one line? Professor Conrey's research is centered on the study of the zeros of zeta-functions. As part of this project, Professor Conrey will also continue his work with Math Teachers' Circles, which are a collection of 39 problem solving groups all across the country that involve professional mathematicians and Middle School math teachers working together to build communities of problem solvers.