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.
