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$$p$$-adic zeta functions at $$s=0$$. (Fonctions zêta $$p$$-adiques en $$s=0$$.) (French) Zbl 0864.11062
Let $$K$$ be a number field, and let $$S$$ denote a finite set of places containing all the archimedean ones. Let $${\mathcal O}_{K,S}$$ be the ring of $$S$$-integers, $$h({\mathcal O}_{K,S})$$ the class number of $${\mathcal O}_{K,S}$$, and $$R_\infty ({\mathcal O}^\times_{K,S})$$ the regulator. Let $$\zeta_{K, \infty, S}$$ denote the (incomplete) Dedekind zeta function of $$K$$. It is well known [see: J. Tate, Les conjectures de Stark sur les fonctions $$L$$ d’Artin en $$s=0$$, Prog. Math. 47 (1984; Zbl 0545.12009)] that near $$s=0$$ we have $\zeta_{K,\infty,S} (s) \sim- h({\mathcal O}_{K, S}) R_\infty ({\mathcal O}^\times_{K,S}) s^{|S|-1}.$ It is the purpose of this article to prove a $$p$$-adic analogue of the above formula (Theorem 6). The main ingredient in the proof is Lemma 8, which gives the Taylor expansion of a certain $$\mathbb{C}_p$$-valued analytic function of several variables (and in particular, the first non-trivial term for the Taylor expansion of the Katz $$p$$-adic $$L$$-function at the trivial character). From this the result follows by a short calculation (Lemma 9).

##### MSC:
 11S40 Zeta functions and $$L$$-functions 11R42 Zeta functions and $$L$$-functions of number fields
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