Irrationality measures and transcendence in positive characteristic
type: Item
instance of: scholarly article
title: Irrationality measures and transcendence in positive characteristic
zbMATH Open document ID: 0832.11022
author: Bernard de Mathan
published in: Journal of Number Theory
publication date: 1996-03-07T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11J81
zbMATH DE Number: 809110
zbMATH Keywords: special zeta values
zbMATH Keywords: Carlitz module
zbMATH Keywords: irrationality measure
zbMATH Keywords: transcendence
reviewed by: David Goss
MaRDI profile type: Publication
full work available at URL: jnth.1995.1104
OpenAlex ID: W2084163968
DOI: 10.1006/JNTH.1995.1104
description: scientific article; zbMATH DE number 809110