Irrationality measures and transcendence in positive characteristic
Let \(\mathbb{F}_r\) be the finite field with \(r\) elements. Let \({\mathbf A}:= \mathbb{F}_r [T]\), \({\mathbf k}:= \mathbb{F}_r (T)\) and \({\mathbf K}:= \mathbb{F}_r ((1/T))\). Let \(j\) be a positive integer and set \(\zeta_{\mathbf A} (j):= \sum_{\substack{ n\in {\mathbf A}\\ n\text{ monic} }} n^{-j}\). These zeta- values were introduced long ago by L. Carlitz and have many analogous properties to the values of the Riemann zeta function at the positive integers. Among them is an Euler-type formula (originally discovered by Carlitz) which can be stated as follows. There exists a transcendental element \(\xi\) (the period of the Carlitz module) such that for those \(j\) with \((r- 1)\mid j\) we have \(\zeta_{\mathbf A} (j)/ \xi^j \in {\mathbf k}^*\). Thus for these values of \(j\) (as with even values for the Riemann zeta function) we deduce the transcendence over \({\mathbf k}\) of \(\zeta_{\mathbf A} (j)\). Building on the work of Anderson and Thakur on special points of the \(n\)- th tensor power of the Carlitz module, \textit{J. Yu} [Ann. Math., II. Ser. 134, 1-23 (1991; Zbl 0734.11040)]\ established that \(\zeta_{\mathbf A} (j)\) is transcendental for all positive \(j\). Moreover, he showed that for those \(j\) not divisible by \(r-1\), the quotient \(\zeta_{\mathbf A} (j)/ \xi^j\) is still transcendental over \({\mathbf k}\). In the paper being reviewed the author presents a new and elementary proof of this latter result for some values of \(j\). This proof is related to the notion of irrationality measure and also gives the transcendence of some polynomial combinations of these elements.
scientific article; zbMATH DE number 809110