Dirichlet Series Expansions of p-adic L-Functions

TL;DR

This paper establishes Dirichlet series expansions for p-adic L-functions associated with Dirichlet characters, using transformations and p-adic measures, with explicit formulas and simplified cases.

Contribution

It introduces a new Dirichlet series expansion for p-adic L-functions for regularization parameters prime to p and the conductor, including explicit formulas and alternative proofs.

Findings

Dirichlet series expansion for p-adic L-functions with regularization parameter c

Explicit formulas for values of regularized Bernoulli distribution

Simplified expansion for c=2 case

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.