On the irrationality measure for certain series
type: Item
instance of: scholarly article
title: On the irrationality measure for certain series
zbMATH Open document ID: 1185.11045
author: Kenji Amano
author: Yohei Tachiya
published in: Monatshefte für Mathematik
publication date: 2009-03-24T00:00:00Z
review text: The main result of this paper is following. Theorem: Let \(K\) be either the rational number field \( \mathbb Q\) or an imaginary quadratic field. Let \(q\) be an integer with \(q\in K\) and \(\mid q\mid >1\). Assume that \(r\in K^\times\) such that \(\mid r\mid <\mid q\mid \). Suppose that \(l\) is an integer with \(1\leq l\). Let \(q^n\not= r^l\) for all \(n=1,2,\cdots\). Then \(\alpha =\sum_{n=1}^\infty \frac{r^n}{q^n-r^l}\notin K\). Further, for any \(\varepsilon >0\) the inequality \(\mid \alpha -\frac pq\mid \geq \frac 1{\mid q\mid^{6l^2+\varepsilon}}\) holds for all integers \(p,q\in K\) with the sufficiently large \(\mid q\mid\). The proof makes use the special integral of the Borwein's type.
reviewed by: Jaroslav Hančl
Mathematics Subject Classification ID: 11J82
Mathematics Subject Classification ID: 11J72
zbMATH DE Number: 5532496
zbMATH Keywords: irrationality
zbMATH Keywords: measure of irrationality
zbMATH Keywords: infinite series
MaRDI profile type: Publication
full work available at URL: s00605-008-0545-1
OpenAlex ID: W2039162304
cites work: Linear independence of the values of transcendental solutions of certain functional equations.
cites work: On the irrationality of certain series
cites work: Linear independence of certain Lambert and allied series
cites work: On approximation measures of q-logarithms
cites work: New irrationality measures for 𝑞-logarithms
cites work: Modular functions and transcendence questions
cites work: Irrationality of certain Lambert series
cites work: Heine's basic transform and a permutation group for q-harmonic series
DOI: 10.1007/S00605-008-0545-1
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description: scientific article; zbMATH DE number 5532496