A Dirichlet series expansion for the p-adic zeta-function

J. Aust. Math. Soc. 81 (2006), 215-224

A Dirichlet series expansion for the p-adic zeta-function

Daniel Delbourgo
Department of Mathematics
University Park
Nottingham
England NG7 2RD
dd@maths.nott.ac.uk

Abstract

We prove that the p-adic zeta-function constructed by Kubota and Leopoldt has the Dirichlet series expansion

$\zeta_p(k,\omega^{1-k}) = \frac{1}{(2-{4}\cdot{2^{-k}})} \sum_{n=1}^{\infty} \sum_{\substack{m=p^{n-1}\\ p\nmid m}}^{p^n} \frac{(-1)^{m+1}}{m^k} \quad\text{at all }\ k\in \mathbb Z,$

where the convergence of the first summation is for the p-adic topology. The proof of this formula relates the values of $\zeta_p(-s,\omega^{1+\beta})$ for $s\in\mathbb Z_p$ , with a branch of the ` $s^{\text{th}}$ -fractional derivative' of a suitable generating function.

Download the article in PDF format (size 81 Kb)

Australian Mathematical Publishing Association Inc.