Linear combinations of \(\zeta(s)/ \Pi^ s\) over \(F_ q(x)\) for \(1\leq s\leq q-2\)
Let \(k= F_q (x)\) be the rational function field over the finite field with \(q\) elements. The author proves that \(k\)-linear combinations of the values \(\zeta (s)/ \Pi^s\), for \(1\leq s\leq q-2\) and \(q\neq 2\), are transcendental over \(k\). Here \(\zeta (s)\) is the Carlitz zeta function for \(k\) and \(\Pi\) is the period of the Carlitz module. The proof in the paper is an application of a theorem of \textit{G. Christol}, \textit{T. Kamae}, \textit{M. Méndes-France}, and \textit{G. Rauzy} [Bull. Soc. Math. Fr. 108, 401-419 (1980; Zbl 0472.10035)] which gives a criterion derived from the theory of automata for a formal series \(\sum a_n x^{-n}\in F_q ((1/x))\) to be algebraic over \(F_q (x)\). See also the author's earlier paper [J. Théor. Nombres Bordx. 5, No. 1, 53-77 (1993; Zbl 0784.11025)] which considers the transcendence of the individual values \(\zeta (s)/ \Pi^s\) from a similar point of view, and work of \textit{J. Yu} [Ann. Math., II. Ser. 134, 1-23 (1991; Zbl 0734.11040)] which gives general results on transcendence of special values in the function field setting.
scientific article; zbMATH DE number 798702