On the measure of irrationality of the number \(\pi \)
type: Item
instance of: scholarly article
title: On the measure of irrationality of the number \(\pi \)
zbMATH Open document ID: 1257.11072
DOI: 10.1134/S0001434610090294
author: V. Kh. Salikhov
published in: Mathematical Notes
publication date: 2011-11-25T00:00:00Z
review text: The author proves that for all \( p,q\in\mathbb N\), \(q\geq q_0\), the inequality \(|\pi- p/q|> q^{-\nu}\) (where \(\nu=7.606308\dots\)) holds, and so gives a new estimate of the measure of irrationality of the number \(\pi\). This result improves the previous record \(\nu= 8.016045\dots\) by \textit{M. Hata} [Acta Arith. 63, No. 4, 335--349 (1993; Zbl 0776.11033)]. In order to prove this result, the author considers the integral \[ J= (1/i) \int^{4+2i}_{4-2i} R(x)\,dx, \] where \[ R(x)= \sum^{5n-2}_{i=0} b_i x^i+ \sum^{5n}_{i=0} (a_i x^{-i-1}+ a_i(10- x)^{-i-1}), \] and uses the saddle-point method.
Mathematics Subject Classification ID: 11J82
Mathematics Subject Classification ID: 11J72
zbMATH DE Number: 5980699
zbMATH Keywords: measure of irrationality
zbMATH Keywords: approximation by rational fractions
zbMATH Keywords: partial-fraction expansion
zbMATH Keywords: Leibniz formula
zbMATH Keywords: linear form
reviewed by: Yaochen Zhu
MaRDI profile type: Publication
cites work: Q5824734
cites work: Q4774099
cites work: Legendre type polynomials and irrationality measures.
cites work: A lower bound for rational approximations to \(\pi\)
cites work: Rational approximations to π and some other numbers
cites work: On the irrationality measure of $ \pi$
full work available at URL: s0001434610090294
OpenAlex ID: W1968531545
description: scientific article; zbMATH DE number 5980699