Abstract
Zeta functions and zeta values play a central role in Modern Number Theory and are connected to practical applications in codes and cryptography. The significance of these objects is demonstrated by the fact that two of the seven Clay Mathematics Million Dollar Millennium Problems are related to these objects, namely the Riemann hypothesis and the Birch and Swinnerton-Dyer conjecture. We first recall results and well-known conjectures concerning these objects over number fields. If time permits, we will present recent developments in the setting of function fields. This is a joint work with Im Bo-Hae and Kim Hojin among others.