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.

这个项目的重点是解析数论中的问题，并涉及黎曼ζ函数和更一般的l函数的性质。l -函数是复平面上的函数，它们通常编码有关算术对象的有趣信息，例如素数、类数或椭圆曲线的秩。例如，黎曼ζ函数（它是l函数的一个例子）与计算小于一个大数的素数的数量的问题密切相关。了解l -函数的解析性质，例如它们的零点位置或增长率，通常有助于了解感兴趣的算术问题。该项目的主要目标是提高对某些l函数族性质的认识，并获得算术应用。该项目的教育部分涉及不同阶段的学生群体，从高中生到初级研究人员。在教育活动中，PI将举办以年轻数学家为中心的解析数论暑期学校，并每年在UCI为有才华的高中生举办夏令营。在更技术性的层面上，该项目将通过研究l函数的比率和矩来研究它们的零点。虽然l函数的正矩已经被很好地理解了，但对负矩和负比的了解却少得多，而负矩和负比在该领域的许多难题中都有应用。计划中的研究将使用随机矩阵理论、几何、筛理论和分析的见解。主要目标分为两个主题。第一个主题是建立研究l -函数负矩的一般框架，提出关于负矩的完全猜想并证明部分结果。第二个主题涉及证明关于l函数在特殊点上的新的不消失结果。l函数在特殊点处的值通常携带重要的算术信息；PI计划表明，广泛的l -函数类不会在中心点（即临界带的中心，所有非平凡的零都被推测在那里）消失，以及研究不同l -函数值之间的相关性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。