A Note on Regularized Bernoulli Distributions and P-Adic Dirichlet Expansions

We consider Bernoulli distributions and their regularizations, which are measures on the p-adic integers Zp. 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 Zp that takes the values ±1/2 on clopen balls. We apply this to 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.