CAREER: New directions in the study of zeros and moments of L-functions

职业：L 函数零点和矩研究的新方向

• 批准号: 2339274
• 负责人: Alexandra Florea
• 金额: $ 50万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2029-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2339274&HistoricalAwards=false
• 关键词: CAREER; New; directions; study; zeros

This project focuses on questions in analytic number theory, and concerns properties of the Riemann zeta-function and of more general L-functions. L-functions are functions on the complex plane that often encode interesting information about arithmetic objects, such as prime numbers, class numbers, or ranks of elliptic curves. For example, the Riemann zeta-function (which is one example of an L-function) is closely connected to the question of counting the number of primes less than a large number. Understanding the analytic properties of L-functions, such as the location of their zeros or their rate of growth, often provides insight into arithmetic questions of interest. The main goal of the project is to advance the knowledge of the properties of some families of L-functions and to obtain arithmetic applications. The educational component of the project involves groups of students at different stages, ranging from high school students to beginning researchers. Among the educational activities, the PI will organize a summer school in analytic number theory focusing on young mathematicians, and will run a yearly summer camp at UCI for talented high school students.At a more technical level, the project will investigate zeros of L-functions by studying their ratios and moments. While positive moments of L–functions are relatively well-understood, much less is known about negative moments and ratios, which have applications to many difficult questions in the field. The planned research will use insights from random matrix theory, geometry, sieve theory and analysis. The main goals fall under two themes. The first theme is developing a general framework to study negative moments of L-functions, formulating full conjectures and proving partial results about negative moments. The second theme involves proving new non-vanishing results about L-functions at special points. Values of L-functions at special points often carry important arithmetic information; the PI plans to show that wide classes of L-functions do not vanish at the central point (i.e., the center of the critical strip, where all the non-trivial zeros are conjectured to be), as well as to study correlations between the values of different L-functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目着重于分析数理论中的问题，并涉及Riemann Zeta功能的特性和更一般的L功能。 L功能是复杂平面上的函数，通常编码有关算术对象的有趣信息，例如质数，班级数量或椭圆曲线的等级。例如，Riemann Zeta功能（这是L功能的一个例子）与计算小于大数字的数量数量的问题紧密相关。了解L功能的分析特性，例如其零的位置或其增长率，通常会深入了解感兴趣的算术问题。该项目的主要目标是提高某些L功能家族的特性并获得算术应用的知识。该项目的教育组成部分涉及不同阶段的学生小组，从高中生到开始研究人员。在教育活动中，PI将组织一所分析数理论的暑期学校，重点是年轻的数学家，并将在UCI为才华横溢的高中生举办一个年度夏令营。在更具技术性的水平上，该项目将通过研究其比率和时间来调查L功能的零。尽管L型功能的积极时刻是相对良好的理解，但对负面的时刻和比率知之甚少，这在该领域的许多困难问题上都有应用。计划研究将使用随机矩阵理论，几何，筛理论和分析中的见解。主要目标属于两个主题。第一个主题是开发一个通用框架来研究L功能的负面时刻，提出了充分的猜想并证明了有关负面时刻的部分结果。第二个主题涉及在特殊点证明有关L功能的新的非变化结果。特殊点的L功能值通常具有重要的算术信息； PI计划表明，在中心点（即，关键地带的中心，所有非平凡的零零的猜想都被认为是），以及研究不同的L函数的价值之间的相关性，这是NSF的法定任务和通过评估的构成构成的范围，该奖项在不同的范围内都表现出了良好的影响。