Dirichlet series expansions of p-adic L-functions

Heiko Knospe,Lawrence C. Washington

doi:10.1007/s12188-021-00244-0

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.

Highlights

Let p be a prime, let q = p if p is odd and q = 4 if p = 2, and let be a Dirichlet character of conductor f
It is well known that Lp(s, ) is identically zero for odd . p-adic L-functions have a long history and the primary constructions going back to Kubota-Leopoldt [6] and Iwasawa [3] are via the interpolation of special values of complex L-functions
The expansion is simple for c = 2, and this parameter can be used for p ≠ 2 and Dirichlet characters with odd conductor

Summary

Introduction

For powers of the Teichmüller character of conductor q, one obtains the p-adic zeta functions p,i = Lp(s, 1−i) , where i = 0, 1, ... We derive similar, but slightly different, expansions for p-adic L-functions. 3, we give an explicit formula for the values of E1,c and derive the Dirichlet series expansion from (2). The expansion is simple for c = 2 , and this parameter can be used for p ≠ 2 and Dirichlet characters with odd conductor. For this case we obtain similar results as in [1, 2], and [4].

Expansions of p‐adic L‐functions

Regularized Bernoulli distributions

Expansions for different regularization parameters