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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).

11S40 Zeta functions and \(L\)-functions
11R42 Zeta functions and \(L\)-functions of number fields
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