Dirichlet Series Expansions of p-adic L-Functions

Abstract

We study $p$-adic $L$-functions $L_p(s,\chi )$ for Dirichlet characters $\chi$. We show that $L_p(s,\chi )$ has a Dirichlet series expansion for each regularization parameter $c$ that is prime to $p$ and the conductor of $\chi$. 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 an alternative proof of the expansion using $p$-adic measures and give an explicit formula for the values of the regularized Bernoulli distribution. The result is particularly simple for $c=2$, where we obtain a Dirichlet series expansion that is similar to the complex case.

9 pages; updated Remark 2.7 and some minor changes. Abh. Math. Semin. Univ. Hambg. (2021)