Why Legendre made a wrong guess about \(\pi(x)\), and how Laguerre's continued fraction for the logarithmic integral improved it.
type: Item
instance of: scholarly article
title: Why Legendre made a wrong guess about \(\pi(x)\), and how Laguerre's continued fraction for the logarithmic integral improved it.
zbMATH Open document ID: 1051.11002
DOI: 10.1007/BF02984842
author: Friedrich L. Bauer
published in: The Mathematical Intelligencer
publication date: 2003-12-16T00:00:00Z
review text: This gives an historical account of attempts to approximate \(\pi(x)\). Laguerre's continued fraction for \(li(x)\) shows that convergence to \(\pi(x)\sim x/(\ln x- 1)\) is very slow and Legendre had available calculations of \(\pi(x)\) only for \(x\) up to \(4.10^5\). Tables are given showing the discrepancies between approximations to \(\pi(x)\) by various methods and for various values of \(x\) up to \(10^{22}\). (The largest value of \(x\) for which \(\pi(x)\) is currently known is \(4.10^{22}\)).
reviewed by: H. J. Godwin
Mathematics Subject Classification ID: 11-03
Mathematics Subject Classification ID: 01A65
zbMATH DE Number: 1998264
zbMATH Keywords: approximatoins to \(\pi(x)\)
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02984842
OpenAlex ID: W2462820427
description: scientific article; zbMATH DE number 1998264