This is a presentation I gave for my Algebraic Number Theory class during the Spring 2019 semester. In it I prove the functional equation for the Riemann zeta function for global function fields of transcendence degree one.

Today we’re going to talk about something quite exciting and unexpected: the arithmetic of polynomial rings over finite fields and its similarity to the arithmetic of the integers. We’ll first run though some preliminary observations about the arithmetic of the polynomial rings which will suggest that it behaves similar to the arithmetic of the integers.