On sums of four smooth squares
type: Item
instance of: scholarly article
title: On sums of four smooth squares
zbMATH Open document ID: 0936.11055
author: Gergely Harcos
published in: Journal of Number Theory
publication date: 2000-05-24T00:00:00Z
review text: The author applies the Hardy-Littlewood circle method together with some of his results in [Acta Arith. 80, 165-185 (1997; Zbl 0871.11066)] to continue his investigation of that work. In the present paper he obtains: Let \(F(M)= (\log M \log \log M)^{1/2}\) and \(P(n)\) denote the greatest prime factor of the integer \(n\). Then almost every natural number \(M\) not divisible by 8 has at least \(M \exp(-F(M)/8)\) representations of the form \[ n_1^2+ n_2^2+ n_3^2+ n_4^2= M, \] where \(n_j\) are integers satisfying \[ P(n_1 n_2 n_3 n_4)< \exp(20 F(M)). \] More precisely, the number of exceptions up to \(N\) is \(\ll N\exp (-F(N)/16)\).
reviewed by: Ming-Chit Liu
Mathematics Subject Classification ID: 11P05
Mathematics Subject Classification ID: 11P55
zbMATH DE Number: 1309272
zbMATH Keywords: Waring problem
zbMATH Keywords: quadratic diagonal form
zbMATH Keywords: Hardy-Littlewood circle method
MaRDI profile type: Publication
full work available at URL: jnth.1999.2380
OpenAlex ID: W2040911887
cites work: Q3686786
cites work: Lagrange's Four Squares Theorem with almost prime variables.
cites work: Waring's problem with small prime factors
cites work: Q3261446
cites work: The Hardy-Littlewood method.
DOI: 10.1006/JNTH.1999.2380
description: scientific article; zbMATH DE number 1309272