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.

该奖项属于解析数论领域，重点研究在该领域发挥核心作用的 L 函数。 L 函数是黎曼 zeta 函数的推广，它编码有关素数分布的重要信息。更一般地说，L 函数封装了人们非常感兴趣的重要算术对象的信息（例如椭圆曲线的秩或类编号）。该项目旨在开发 L 函数研究的新工具，特别是关注它们的零点和中心值，以期提取算术信息。研究人员将为博士生提供建议，并为来自数学领域代表性不足的群体的学生提供指导。该奖项将研究家庭中 L 函数的矩和比率，包括数域和函数域设置。最终目标是理解渐进时刻低阶项的工作机制以及每个族的结构。 该项目的另一个主题是研究家庭中 L 函数的零点，重点是获得消失和非消失的结果。所采用的方法将是解析数论技术，包括使用函数方程、求和公式、指数和、筛理论思想和随机矩阵理论启发的见解。该奖项反映了 NSF 的法定使命，并通过使用基金会的评估进行评估，认为值得支持。智力价值和更广泛的影响审查标准。

项目成果

The Ratios Conjecture and upper bounds for negative moments of 𝐿-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: 2023-01
• 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

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: 2021-11
• 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;Wang, Jiuya
• 通讯作者: Wang, Jiuya