Zeros and Moments of L-Functions

该奖项是在解析数论领域，专注于l函数的研究，它在该领域发挥着核心作用。l -函数是黎曼ζ函数的推广，它编码了关于素数分布的重要信息。更一般地说，l -函数封装了有关重要算术对象的信息（例如椭圆曲线的秩或类数），这是非常有趣的。本项目旨在开发l函数研究的新工具，特别是通过关注它们的零点和中心值，以期提取算术信息。研究者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，包括在数域和函数域设置中。最后的目标是理解低阶项在矩渐近中工作的机制和每个族的结构。项目的另一个主题是研究族中l函数的零，重点是得到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，筛子理论思想和随机矩阵理论启发的见解。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。

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.