On the proportion of squarefree numbers among sums of cubic polynomials
type: Item
instance of: scholarly article
title: On the proportion of squarefree numbers among sums of cubic polynomials
zbMATH Open document ID: 1472.11270
author: N. E. Zubov
published in: The Ramanujan Journal
publication date: 2021-05-06T00:00:00Z
review text: \textit{C. Hooley} [Mathematika 14, 21--26 (1967; Zbl 0166.05203)] proved that if \(f(x)\in Z[x]\) is an irreducible polynomial degree \(d\ge 3\) then \(f(n)\) takes \(c_fx+o(x)\) \((d-1)\)-free values of \(n\le x\). \textit{G. Greaves} [Q. J. Math., Oxf. II. Ser. 43, No. 169, 45--65 (1992; Zbl 0768.11034)] obtained an analogue result for binary forms. Assuming the \(abc\)-conjecture, \textit{B. Poonen} [Duke Math. J. 118, No. 2, 353--373 (2003; Zbl 1047.11021)] proved a similar result for squarefree values of multivariable polynomials. In the paper under review, the author gives an asymptotic formula for the proportion of squarefree numbers among \(f_1(x_1)+ f_2(x_2)+\cdots+ f_s(x_s)\) where \(s\ge 3\), each \(f_i\) is a nonzero cubic polynomial with integer coefficients and \(x_i\in Z\), \(a_iB\le x_i\le b_iB\) \((i=1,\dots,s)\), \(B\to\infty\).
reviewed by: Mihály Szalay
Mathematics Subject Classification ID: 11P55
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 11N32
zbMATH DE Number: 7344959
zbMATH Keywords: squarefree numbers
zbMATH Keywords: multivariable polynomials
MaRDI profile type: Publication
full work available at URL: s11139-019-00239-9
OpenAlex ID: W3033281658
cites work: POWER-FREE VALUES OF BINARY FORMS
cites work: On the power free values of polynomials
cites work: Q5587215
cites work: Multiplicative number theory. I. Classical theory
cites work: Squarefree values of multivariable polynomials
cites work: Q4674810
DOI: 10.1007/S11139-019-00239-9
description: scientific article; zbMATH DE number 7344959