# What are the branches of the $p$-adic zeta function?

I'm reading the book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz. In it, Koblitz wants to iterpolate the Riemann Zeta function for the values $\zeta_p(1-k)$ with $k \in \mathbb{N}$, where $\zeta_p(s) = (1-p^s)\zeta(s)$. Doing this, he defines the next function:

$$\zeta_{p,s_0}(s) = \frac{1}{\alpha^{-(s_0+(p-1)s)}-1} \int_{\mathbb{Z}_p^*}x^{s_0+(p-1)s-1}\mu_{1,\alpha},$$ with $s_0 \in \{0,1,2,\ldots,p-2\}$ and $s \in \mathbb{Z}_p$. He then says that this $\zeta_{p,s_0}$ are 'branches' of $\zeta_p$. My question is: how does $\zeta_{p,s_0}$ define a branch ? $\zeta_{p,s_0}$ generally does not interpolate the values $\zeta_p(1-k)$ even for $k \equiv s_0 \bmod (p-1)$, since we have $$k =s_0+k_0(p-1) \Rightarrow \zeta_p(1-k) = \zeta_{p,s_0}(k_0)$$ and not $\zeta_p(1-k) = \zeta_{p,s_0}(1-k)$. So my question is really: what are the real branches of the $p$-adic zeta function ?

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## 1 Answer

According to the idea of Tate's thesis, the Riemann zeta function $\zeta(s)$ should be thought of as a distribution on the idèles $\text{GL}_1(\mathbf A_\mathbf Q)$ of $\mathbf Q$. Integrating this distribution against the complex characters of $\text{GL}_1(\mathbf A_\mathbf Q)$ gives the values of the zeta function.

Iwasawa showed that the $p$-adic Riemann zeta function constructed by Kubota and Leopoldt admits a similar construction. The object which plays the role of the idèles is $\mathbf Z_p^\times$; the role of complex characters of the idèles is played by the $p$-adic characters in the weight space $W=\text{Hom}_{\text{cont}}(\mathbf Z_p^\times, \mathbf Q_p^\times)$ (continuous homomorphisms). This group is canonically isomorphic to $\mathbf Z/(p-1)\mathbf Z \times \mathbf Z_p$. The zeta function can therefore be seen as a $p$-adic valued function on $\mathbf Z/(p-1)\mathbf Z \times \mathbf Z_p$. The various restrictions to $s \in \mathbf Z/(p-1)\mathbf Z$ are the "branches" of the zeta function as you are seeing them in Kobliz's book.

Edit: I have changed this answer, so Sanchez's comment is not related. (What I had before was true but I don't think it was the best way to see this.)

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Can you say a few words on why this is the natural way? That one should look at zeta values as constant term of modular forms is pretty new to me. Does this generalize to special values of other L functions? –  user27126 Feb 26 '13 at 6:02