Zeros and Moments of L-Functions

L 函数的零点和矩

• 批准号: 2101769
• 负责人: Alexandra Florea
• 金额: $ 23.1万
• 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: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101769&HistoricalAwards=false
• 关键词: Zeros; Moments; Functions

This award is in the area of analytic number theory, focusing on the study of L-functions, which play a central role in the field. L-functions are generalizations of the Riemann zeta function, which encodes important information about the distribution of prime numbers. More generally, L-functions package information about important arithmetic objects (such as the rank of an elliptic curve or the class number) which are of great interest. This project aims to develop new tools in the study of L-functions, particularly by focusing on their zeros and their central values, with the hope of extracting arithmetic information. The investigator will advise PhD students and provide mentorship to students coming from groups that are underrepresented in mathematics.The award will study moments and ratios of L-functions in families, both in the number field and in the function field setting. The final goal is that of understanding the mechanism through which lower-order terms in the moment asymptotics work and the structure of each family. Another theme of the project is studying zeros of L-functions in families, focusing on obtaining vanishing and non-vanishing results. The methods employed will be analytic number theory techniques, including use of functional equations, summation formulas, exponential sums, sieve theory ideas and random matrix theory inspired insights.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.

该奖项是在分析数理论领域，重点是对L功能的研究，在该领域中起着核心作用。 l功能是Riemann Zeta函数的概括，该功能编码了有关素数分布的重要信息。更一般而言，L-功能包装的信息有关重要的算术对象（例如椭圆曲线的等级或班级编号的等级）。该项目旨在在研究L功能的研究中开发新的工具，尤其是通过关注其零和其中心价值，希望提取算术信息。调查人员将为博士生提供建议，并为来自数学中人数不足的小组的学生提供指导。该奖项将在数量领域和功能领域设置中研究家庭中L功能的时刻和比率。最终的目标是了解渐变学中低阶项起作用和每个家庭结构的机制。 该项目的另一个主题是研究家庭中L功能的零，重点是获得消失和不变结果。所采用的方法将是分析数理论技术，包括使用功能方程，求和公式，指数总和，筛理理论思想和随机矩阵理论启发了洞察力。该奖项反映了NSF的法定任务，并被认为是通过基金会的评估来通过评估来支持的。智力优点和更广泛的影响审查标准。

项目成果

Hilbert transforms and the equidistribution of zeros of polynomials

希尔伯特变换和多项式零点的均分布

• DOI: 10.1016/j.jfa.2021.109199
• 发表时间: 2021
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: Carneiro, Emanuel;Das, Mithun Kumar;Florea, Alexandra;Kumchev, Angel V.;Malik, Amita;Milinovich, Micah B.;Turnage-Butterbaugh, Caroline;Wang, Jiuya
• 通讯作者: Wang, Jiuya

The Ratios Conjecture and upper bounds for negative moments of ?-functions over function fields

函数域上 ?-函数负矩的比率猜想和上限

• DOI: 10.1090/tran/8907
• 发表时间: 2023
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Bui, Hung;Florea, Alexandra;Keating, Jonathan
• 通讯作者: Keating, Jonathan