CAREER: New directions in the study of zeros and moments of L-functions

本课题主要研究解析数论中的问题，并研究Riemann Zeta函数和更一般的L函数的性质。L函数是复平面上的函数，通常编码关于算术对象的有趣信息，如素数、类号或椭圆曲线的秩数。例如，黎曼Zeta函数(这是L函数的一个例子)与计算小于大数的素数的问题密切相关。了解L函数的分析性质，如它们的零点位置或增长率，通常有助于深入了解感兴趣的算术问题。该项目的主要目的是增进对某些L函数族的性质的了解，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将组织一个面向年轻数学家的解析数论暑期学校，并将在加州大学洛杉矶分校为有才华的高中生举办一年一度的夏令营。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函数的正矩相对较好地被理解，但对负矩和负比的了解要少得多，它们在该领域的许多困难问题中都有应用。计划中的研究将使用随机矩阵理论、几何、筛子理论和分析的见解。主要目标有两个主题。第一个主题是建立一个研究L函数负矩的一般框架，提出完整的猜想并证明关于负矩的部分结果。第二个主题涉及证明关于L的新的非零化结果--在特殊点上的函数。L函数在特定点的值通常带有重要的算术信息；PI计划证明广泛的L函数类不会在中心点(即临界带的中心，所有非平凡零点都被猜想为中心点)处消失，并研究不同L函数的值之间的相关性。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智力优点和更广泛的影响审查标准进行评估，被认为值得支持。

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.