RUI: Automorphic Summation Formulae

这个项目探索Riemann Zeta函数的矩和自同构形式之间的关系。受最近关于矩的猜想的启发，Beineke教授计划继续发展一类新的自同构求和公式，以便在Riemann Zeta函数的2k阶矩与GL(2k)上的Eisenstein级数之间建立联系。在k=1的情况下的初步结果是由贝内克教授与斯坦福大学的丹尼尔·邦普教授合作开发的。这是数论的一个项目，数论是数学中最古老的分支之一。数论的基础在于对正整数的研究和在这些整数中寻找模式。数论中的一个有趣的函数是Riemann Zeta函数，它可以提供关于素数模式的信息。关于该函数的结果可能在密码学和互联网安全方面有应用。贝内克教授的项目研究了其他被称为自同构形的数论对象，以及它们与Riemann Zeta函数平均值的可能联系。这些联系最初是由随机矩阵理论的结果推动的，随机矩阵理论最初是用来建立实验物理模型的学科。

This project explores the relationship between moments of the Riemann zeta function and automorphic forms. Motivated by recent conjectures on the moments, Professor Beineke plans to continue development of a new class of automorphic summation formulae in order to establish connections between the 2k-th moment of the Riemann zeta function and Eisenstein series on GL(2k). Initial results in the case where k = 1 have been developed by Professor Beineke in collaboration with Professor Daniel Bump at Stanford University.This is a project in number theory, one of the oldest branches of mathematics. The foundations of number theory lie in the study of the positive integers and finding patterns in these integers. A function of interest in number theory, which may provide information about patterns of prime numbers, is the Riemann zeta function. Results about this function could have applications to cryptography and Internet security. Professor Beineke's project investigates other number-theoretic objects called automorphic forms, and their possible connections to average values of the Riemann zeta function. These connections were initially motivated by results stemming from Random Matrix Theory, a subject originally used to develop models in experimental physics.