Families of L-Functions and Analytic Number Theory

这个研究项目以L函数为中心，它是一种数学对象，用于研究数论中的一系列问题。L函数是一种特殊的函数，它把研究一个算术对象的信息集中在一起，对每个素数p取模p。例如，Riemann Zeta函数就是L函数的最简单的例子，在研究素数的分布时是不可或缺的。其他类型的L函数对于了解某些多项式方程是否有解，或者更广泛地说，有多少解是至关重要的。这些问题往往与L在某一特殊时刻的作用有多大有关。这个项目的大部分内容是关于开发新的工具来研究L函数如何适应家庭，并使用这些工具来更好地理解个体L函数。这位调查员将继续为博士生提供建议，并与本科生进行指导和合作，特别是通过德克萨斯农工大学本科生研究体验项目。这种类型的指导对于学生为研究生学习做好准备是非常宝贵的，特别是对于来自非博士学位授予机构的本科生以及在STEM领域代表性不足的群体。该项目将研究新的L函数族，并将其用作估计临界线上的L函数的工具。特别是，这项工作的目的是发展新的大筛子不等式，这是在L函数的分析问题中广泛使用的灵活工具。这位研究人员和他的学生将研究L的时刻--比以前考虑的更小的子家族中的函数。另一项工作涉及量子唯一遍历问题的新变体，该问题将L函数族与自同构形式的性质联系起来。研究人员将和他的本科生一起研究广义Dedekind和的性质。所使用的方法将是来自解析数论的技术，如求和公式、函数方程、指数和和积分，以及自同构形式的谱理论，包括Arthur-Selberg迹公式和相对迹公式。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This research project centers on L-functions, which are mathematical objects useful for studying a broad array of questions in number theory. An L-function is a special kind of function that packages together information about an arithmetical object that arises from studying it modulo p for each prime p. For instance, the Riemann zeta function is the simplest example of an L-function and has proven to be indispensable in studying the distribution of the prime numbers. Other types of L-functions are crucial for understanding if certain polynomial equations have solutions, or more generally, how many solutions there are. Often these questions are related to how large the L-function is at a special point. Much of this project concerns the development of new tools for studying how L-functions may fit into families and using these tools to better understand individual L-functions. The investigator will continue to advise PhD students and to mentor and collaborate with undergraduate students, especially through the Texas A&M Research Experience for Undergraduates. This type of mentorship is invaluable in preparing students for graduate studies, particularly for undergraduate students from non-PhD granting institutions as well as from population groups underrepresented in STEM fields. The project will study new families of L-functions and use them as tools for estimating L-functions on the critical line. In particular, the work aims to develop new large sieve inequalities, which are flexible tools broadly useful in analytic problems on L-functions. The investigator and his students will study moments of L-functions in smaller sub-families than have previously been considered. Another line of work concerns new variants on the quantum unique ergodicity problem, which connects families of L-functions to properties of automorphic forms. With his undergraduate students, the investigator will study properties of generalized Dedekind sums. The methods employed will be techniques from analytic number theory such as summation formulas, functional equations, exponential sums and integrals, and the spectral theory of automorphic forms, including the Arthur-Selberg trace formula and the relative trace formula.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.