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
  England NG7 2RD

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. ©  Australian MS