p-adic L-functions and Bernoulli Numbers

Abstract

In this chapter we shall construct p-adic analogues of Dirichlet L-functions. Since the usual series for these functions do not converge p-adically, we must resort to another procedure. The values of $L\left( {s,\chi } \right)$ at negative integers are algebraic, hence may be regarded as lying in an extension of ${\mathbb{Q}_p}$. We therefore look for a p-adic function which agrees with $L\left( {s,\chi } \right)$ at the negative integers. With a few minor modifications, this is possible.