On conditional irrationality measures for values of the digamma function
type: Item
instance of: scholarly article
title: On conditional irrationality measures for values of the digamma function
zbMATH Open document ID: 1110.11021
published in: Journal of Number Theory
publication date: 2007-02-26T00:00:00Z
review text: For \(\theta\in\mathbb{R}\), let \(\beta=\beta(\theta)\in\mathbb{R}\) be the least number with the property that for any \(\varepsilon> 0\) there is \(q_0(\varepsilon> 0\) such that \(|\theta- p/q|>1/(\beta+ \varepsilon)^q\) for all \(p,q\in\mathbb{Z}\) with \(q\geq q_0(\varepsilon)\), and call it the irrationality base of \(\theta\). The authors give several irrationality criteria. One of them is as follows: Denote \(R_n= A_n\theta- B_n\) \((n= 1,2,\dots)\) for some \(A_n,B_n\in\mathbb{Z}\). Let \(\limsup_{n\to\infty}\log|A_n|/n\leq \sigma\), \(\sigma\geq 0\), and \(\lim_{n\to\infty}|nRn|=\tau\), \(\tau\geq 0\). (If \(\tau= 0\), suppose also that \(R_n\neq 0\) for all \(n\geq n_0\)) Then \(\theta\) is irrational and \(\beta(\theta)\leq e^{\sigma r}\). By using these criteria the authors obtain conditional irrationality measures for values at rational numbers of the digamma function \({\Gamma'\over\Gamma}(z)\).
reviewed by: Yaochen Zhu
Mathematics Subject Classification ID: 11J82
zbMATH DE Number: 5129757
zbMATH Keywords: irrationality measure
zbMATH Keywords: irrationality base
zbMATH Keywords: digamma function
zbMATH Keywords: Euler's constant
zbMATH Keywords: Diophantine approximations
author: Kh. Hessami Pilehrood
author: T. Hessami Pilehrood
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.07.008
OpenAlex ID: W2058080101
cites work: Q3859868
cites work: Q3667344
cites work: Rational approximations to π and some other numbers
cites work: Criteria for irrationality of generalized Euler's constant
DOI: 10.1016/J.JNT.2006.07.008
description: scientific article; zbMATH DE number 5129757