L-functions and Multiplicative Number Theory

L 函数和乘法数论

• 批准号: 0100414
• 负责人: Kannan Soundararajan
• 金额: $ 12.43万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-06-01 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100414&HistoricalAwards=false
• 关键词: functions; Multiplicative; Number; Theory

The investigator and his colleagues study various problems inthe analytic theory of L-functions and mean values of multiplicativefunctions. In particular the investigator aims for an improvedunderstanding of the distribution of zeros of L-functions,their behaviour at special values, and the size of extreme valuesin the critical strip. The second main aim of this project isto describe the spectrum of possible mean-values of multiplicativefunctions whose values at primes are constrained to lie in specialsubsets of the unit disc. The investigator, in collaborationwith Granville, intends to develop further the close connection betweensuch mean values and solutions to a family of integral equations.This project concerns problems in the area of multiplicativenumber theory. A fundamental problem in this area is to understandthe zeros of the Riemann zeta function and other L-functions. Thesefunctions encode much information about prime numbers, andother arithmetic objects. Besides being of intrinsic interest tomathematicians, ideas emerging from these problems have found usein computer science, cryptography and coding theory.

研究人员和他的同事研究了L分析理论中的各种问题--函数和乘法函数的平均值。特别是，作者的目的是更好地理解L函数的零点的分布，它们在特定值上的行为，以及在临界带上极值的大小。这个项目的第二个主要目的是描述在素数处的值被限制在单位圆盘的特殊子集内的乘法函数的可能平均值的谱。研究人员与Granville合作，打算进一步发展这些平均值与一族积分方程解之间的密切联系。这个项目涉及乘数理论领域的问题。这方面的一个基本问题是理解黎曼-Zeta函数和其他L函数的零点。这些函数编码关于质数和其他算术对象的大量信息。除了对数学家有内在的兴趣外，从这些问题中产生的想法也被用于计算机科学、密码学和编码理论。