A note on regularized Bernoulli distributions and p-adic Dirichlet expansions

TL;DR

This paper explores regularized Bernoulli distributions on p-adic integers, revealing a simple measure for p>2 and deriving Dirichlet series expansions for p-adic L-functions, connecting p-adic measures with classical Dirichlet series.

Contribution

It introduces a simplified measure for regularized Bernoulli distributions on dic integers for p>2 and applies it to derive Dirichlet series expansions for p-adic L-functions.

Findings

A simple measure taking values bd bd bd bd bd bd bd bd on bdp-adic integers for p>2.

Derivation of Dirichlet series expansions for p-adic L-functions similar to the complex case.

Connection of p-adic measures with classical Dirichlet series studied by Delbourgo.

Abstract

We consider Bernoulli distributions and their regularizations, which are measures on the $p$-adic integers $\mathbb{Z}_p$. It is well known that their Mellin transform can be used to define $p$-adic $L$-functions. We show that for $p>2$ one of the regularized Bernoulli distributions is particularly simple and equal to a measure on $\mathbb{Z}_p$ that takes the values $\pm \frac{1}{2}$ on clopen balls. We apply this to $p$-adic $L$-functions for Dirichlet characters of $p$-power conductor and obtain Dirichlet series expansions similar to the complex case. Such expansions were studied by D. Delbourgo, and this contribution provides an approach via $p$-adic measures.