Diophantine properties of numbers related to Catalan's constant
type: Item
instance of: scholarly article
title: Diophantine properties of numbers related to Catalan's constant
zbMATH Open document ID: 1028.11046
published in: Mathematische Annalen
publication date: 2003-09-16T00:00:00Z
full work available at URL: http://hdl.handle.net/1959.13/803688
review text: The main subject of this paper is to study the arithmetic nature of numbers related to Dirichlet's \(\beta\)-function: \[ \beta(s)=\sum_{n=0}^\infty {(-1)^n\over (2n+1)^s}. \] Lindemann's theorem implies that \(\beta(2n+1)\) is transcendental for \(n\in \mathbb N\). The authors consider the case of even integers and prove: Theorem. Let \(a\) be an even positive integer and let \(\delta(a)\) be the dimension of the \(\mathbb Q\)-vector space spanned by the numbers 1, \(\beta(2)\), \(\beta(4)\), \dots, \(\beta(a)\), then \[ \delta(a) \geq {1+o(1)\over 2+\log(2)} \log (a) \qquad \text{ as} \;a \to \infty. \] Their method is effective and they show that at least one of the numbers \(\beta(2)\), \(\beta(4)\), \dots, \(\beta(14)\) is irrational. The main tool in the proof is the study of well-chosen hypergeometric functions. This study is quite delicate and detailed. The proof is about 15 pages long.
reviewed by: Maurice Mignotte
Mathematics Subject Classification ID: 11J72
zbMATH DE Number: 1982589
zbMATH Keywords: Catalan's constant
zbMATH Keywords: irrationality
author: T. Rivoal
author: Wadim Zudilin
MaRDI profile type: Publication
OpenAlex ID: W1986915956
DOI: 10.1007/S00208-003-0420-2
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description: scientific article; zbMATH DE number 1982589