Analytic Theory of L-functions

L-函数的解析理论

• 批准号: 1101774
• 负责人: John Conrey
• 金额: $ 12万
• 依托单位: American Institute of Mathematics
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101774&HistoricalAwards=false
• 关键词: Analytic; Theory; functions

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.

l功能是编码算术信息的分析数理论中的基本对象。最简单的例子是Riemann Zeta功能，Dirichlet L功能以及与模块化形式相关的L功能。了解这些L功能的价值和零的统计行为是Conrey博士研究的主要主题。他和他的合作者提出了各种项目，涉及L功能的某些方面。一个涉及一个新公式，具有除数函数d（n）的新公式，该公式概括了经典的Voronoi公式，并在Nyman-Beburling方法中应用于Riemann假设；康里（Conrey）将开发进一步的应用程序，以了解矩公式和理解低度功能的问题。康里（Conrey）将使用渐近的大筛（用iwaniec和soundararajan发明）来证明所有dirichlet l功能的统一正方形的猜想乘以任意的dirichlet多项式。第三个项目涉及马祖尔的猜想，即与椭圆曲线相关的对称功率功能很少在其中心点消失。专业康雷（Conrey）的研究在数字理论领域。现代数字理论具有令人惊讶的多样化应用程序，从实现安全的互联网交易，到最佳网络的构建，甚至到对低维拓扑的研究中各种类型的物体进行分类的问题。数字理论家发明的最成功的工具之一是zeta功能。它的最初目的是帮助研究质数。现在，它及其类似物在数量理论上无处不在。但是，Zeta功能仍然有一些非常基本的特性，我们不了解，如果我们确实会带来很大进步。主要的问题是，为什么所有的Zeta功能的零仅在一行上出现？ Conrey教授的研究集中在Zeta功能的零零的研究上。作为该项目的一部分，Conrey教授还将继续与数学老师圈子的合作，这是全国各地39个解决问题的集团的集合，其中涉及专业的数学家和中学数学老师，共同努力建立问题解决者社区。