The distribution of square-free numbers of the form \([n^c]\)
type: Item
instance of: scholarly article
title: The distribution of square-free numbers of the form \([n^c]\)
zbMATH Open document ID: 0926.11066
author: Xiaodong Cao
author: Wenguang Zhai
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1999-06-23T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93718
full work available at URL: jtnb10-2.html
review text: Let \(S_c(x)\) denote the number of positive integers \(n\) not exceeding \(x\) such that \([n^c]\) are square-free. The authors prove an asymptotic representation for \(S_c(x)\) with an explicit expression for the error term, provided that \(1< c< 2\). Now it is possible to use the method of exponential sums. It is shown that the sequence \([n^c]\) \((n=1,2,\dots)\) contains an infinite number of square-free integer whenever \(1< c<61/36\). This improves earlier results.
reviewed by: Ekkehard Krätzel
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 1307386
zbMATH Keywords: distribution of square-free integers
zbMATH Keywords: exponential sums
zbMATH Keywords: exponent pair
MaRDI profile type: Publication
OpenAlex ID: W1996122521
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DOI: 10.5802/JTNB.229
description: scientific article; zbMATH DE number 1307386