Mysterious unramified zeta functions

Project leader: Professor Jayce R. Getz
Project manager: Chung-Ru Lee
Team members: Lucas Fagan, Craig Fiedorek, Diego Sosa-Fundora, Tony Sun, Henry Zhang

Project summary:Mysterious unramified zeta functions
Zeta functions are a marriage of algebra and analysis that are of fundamental importance throughout number theory. One property is that they can often be viewed as Euler products, in particular they admit an infinite product expansion indexed by the prime numbers (including infinity). Professor Getz has recently uncovered a family of zeta functions that are, for the moment, mysterious. Our project investigated a local factor of one of these functions at unramified primes. We used techniques from finite group theory and the combinatorics of symmetric functions to simplify an infinite sum into a rational expression. Obtaining this rational expression form is important to understanding the L-function because it conveys information about arithmetic properties along with zeroes and poles in a way that can not be seen from an infinite sum.