Carlitz \(\zeta\)-function and automata
The author gives a new proof of the transcendence of the values \(\zeta(s)/\Pi^ s\) for integers \(1\leq s\leq q-2\), where \(\zeta(s)\) is the Carlitz zeta function for the rational function field over \(\mathbb{F}_ q\) and \(\Pi\) is the analogue for the Carlitz module of \(\pi\). The transcendece of these zeta values is a special case of Yu's general theorem, which gives the transcendence of \(\zeta(s)/\Pi^ s\) for all integers \(s\) which are not divisible by \(q-1\) [\textit{J. Yu}, Ann. Math. 134, 1-23 (1991; Zbl 0734.11040)]. The proof in the paper is based on a theorem of \textit{G. Christol}, \textit{T. Kamae}, \textit{M. Mendès-France} and \textit{G. Rauzy} [Bull. Soc. Math. Fr. 108, 401-419 (1980; Zbl 0472.10035)] which gives a criterion for a formal series \(\sum a_ n x^{-n}\in \mathbb{F}_ q((1/x))\) to be algebraic over \(\mathbb{F}_ q(x)\).
scientific article; zbMATH DE number 484161
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