Measures and Series Expansions of p-adic L-Functions for Dirichlet Characters

Abstract

This is joint work with Lawrence C. Washington. We study p-adic L-functions $L_p(s,\chi)$ for Dirichlet characters $\chi$. The primary constructions of p-adic L-functions are via the interpolation of special values of complex L-functions. We show that $L_p(s,\chi)$ also has a Dirichlet series expansion for each regularization parameter c that is prime to p and the conductor of the character. The expansion is proved by transforming a known formula for p-adic L-functions and by controlling the limiting behavior. A finite number of Euler factors can be factored off in a natural manner from the p-adic Dirichlet series. We also provide a proof of the expansion using p-adic measures and give an explicit formula for the values of the regularized Bernoulli distribution. The measure and the Dirichlet series expansion is particularly simple for c=2, where we obtain a formula that is similar to the complex case.