**Ralph Greenberg**

University of Washington, USA

July 12, 2012

**The $p$-adic $L$-functions of Kubota and Leopoldt**:
A good part of this talk will be an introduction to the topic. We will discuss the $p$-adic analogues of the Riemann zeta function
and Dirichlet $L$-functions which were constructed in the early 1960s by Kubota and Leopoldt. We also want to discuss a new proof of an old formula proved by Bruce Ferrero and myself in the 1970s. That formula gives the value of the derivative for a Kubota-Leopoldt $p$-adic $L$-function at $s=0$ when the function itself vanishes at that point. The new proof is a joint project with Benjamin Lundell and Shaowei Zhang and relies on studying a certain anaytic function of two $p$-adic variables.