Let \(|\cdot |\) be the classical absolute value defined on the field \(\mathbb F_q(T)\) by \(| P/Q| =q^{\deg P-\deg Q}\) when \(P,Q\) belong to \(\mathbb F_q[T]\). Let \(\mathbb F_q((T))\) be the completion of \(\mathbb F_q(T)\) (i.e. the field of Laurent series in the form \(\sum_{-\infty}^t a_sT^s\) \((t\in\mathbb Z))\). Let \(\mathcal P\) be the set of monic polynomials in \(\mathbb F_q[T]\). Carlitz’ zeta function is then defined from \(\mathbb N^*\) to \(\mathbb F_q((T))\) by \(\zeta(s)=\sum_{f\in\mathcal P} f^{-s}\) (this series is clearly convergent for each \(s\in\mathbb N^*)\). In 1935, L. Carlitz [Duke Math. J. 1, 137–168 (1935; Zbl 0012.04904, JFM 61.0127.01)] showed there exists \(\pi \in\mathbb F_q((T))\) such that, for every \(s\) multiple of \(q-1\), \(\zeta_s\) factorizes in the form \(c_s\pi^s\) with \(c_s\in\mathbb F_q(T)\), and \(\pi\) is transcendental over \(\mathbb F_q(T)\). Here, when \(q=2\), the authors give a measure of irrationality for \(\pi\), hence for \(\zeta(1)\), in taking inspiration from Apéry’s method for convergence acceleration.
For example, they show that for every \(\delta >13/6\) there is at most a finite number of couples \((P,Q)\in\mathbb F_q[T]^2\) such that \[ | \zeta(1)-P/Q| \leq | Q|^{-\delta}. \]
However there are infinitely many couples \((P,Q)\) such that \[ | \zeta (1)-P/Q| \leq | Q|^{-2} 2^{-2\sqrt{\Log_2| Q|}}, \] so that the previous result is not far from being the best possible.