Analytic Theory of L-functions

L-函数的解析理论

• 批准号: 0801264
• 负责人: John Conrey
• 金额: $ 9.53万
• 依托单位: American Institute of Mathematics
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801264&HistoricalAwards=false
• 关键词: Analytic; Theory; functions

L-functions are the basic functions of number theory and are fundamental in the study of prime numbers, solutions to equations in whole numbers, and the uniformity of the distribution of arithmetic sequences. The first L-function is the Riemann zeta-function. The location of its zeros is the subject of the Riemann Hypothesis which is widely regarded as the most important unsolved problem in all of mathmatics. In this project the PI will study various approaches to the Riemann Hypothesis. In addition, he will investigate some very specific statistical properties of the Riemann zeta-function and of families of L-functions in order to more fully understand thes mysterious functions.The PI and his collaborators intend to prove that most of the zeros of Dirichlet L-functions are on the critical line. They also will to develop a general tool, the asymptotic large sieve, to address problems involving averages over all primitive Dirichlet characters of modulus less than a given parameter. They will use this to investigate the spacings between zeros of Dirichlet L-functions. Another project is to determine (conjecturally) the arithemetic part of the distribution of spacings between consecutive zeros of the Riemann zeta-function.Finally, the PI would like to prove the reciprocity formula that he conjectured for Vasyunn sums, and to investigate its relevance in the Nyman-Beurling approach to the Riemann Hypothesis.

L函数是数论的基本函数，在研究素数、整数方程的解和等差数列分布的均匀性方面是基础。第一个L-函数是Riemann zeta-函数。 它的零点的位置是黎曼猜想的主题，它被广泛认为是所有数学中最重要的未解决的问题。在这个项目中，PI将研究黎曼假设的各种方法。此外，他还将研究Riemann zeta函数和L函数族的一些非常具体的统计性质，以便更全面地理解这些神秘的函数。PI和他的合作者打算证明Dirichlet L函数的大多数零点都在临界线上。 他们还将开发一种通用工具，渐近大筛，以解决涉及平均数的所有原始狄利克雷特征模小于一个给定的参数。他们将利用这一点来研究狄利克雷L-函数的零点之间的间距。另一个项目是确定黎曼zeta函数的连续零点之间的间距分布的算术部分。最后，PI想证明他为Vasyunn和所证明的互易公式，并研究其在Nyman-Beurling方法中与黎曼假设的相关性。