Sums of almost equal prime cubes
type: Item
instance of: scholarly article
title: Sums of almost equal prime cubes
zbMATH Open document ID: 1224.11088
author: Sumit K. Garg
published in: Acta Mathematica Hungarica
publication date: 2011-02-08T00:00:00Z
review text: It is known that every sufficiently large integer satisfying the natural congruence conditions is a sum of the cubes of nine primes; and that for sums of 5, 6, 7, or 8 cubes of primes ``almost all'' large integers satisfying the appropriate natural congruences can be represented. The present paper shows that in the latter case one can restrict the primes involved to be almost equal. To be specific, let \(\theta<1/45\) for \(r=5\), or \(<1/30\) for \(r=6\), or \(<1/25\) for \(r=7\), or \(<2/45\) for \(r=8\). Then if \(5\leq r\leq 8\), ``almost all'' numbers \(N\) satisfying the relevant congruence conditions can be written as \(p_1^3+\ldots+p_r^3\) with \[ | p_j-(N/r)^{1/3}| \leq N^{1/3-\theta}. \] The proof uses the circle method. On the minor arcs an estimate for exponential sums over primes in short intervals, due to \textit{J. Liu, G. Lü} and \textit{T. Zhan} [Sci. China, Ser. A 49, 611--619 (2006; Zbl 1106.11033)], plays an important role.
reviewed by: D. R. Heath-Brown
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11P05
Mathematics Subject Classification ID: 11L20
Mathematics Subject Classification ID: 11P55
zbMATH DE Number: 5848590
zbMATH Keywords: circle method
zbMATH Keywords: exponential sum over primes
zbMATH Keywords: short interval
zbMATH Keywords: Goldbach-Waring problem
zbMATH Keywords: cubes
zbMATH Keywords: almost equal
MaRDI profile type: Publication
full work available at URL: s10474-009-9061-6
OpenAlex ID: W1989048025
cites work: Mean values of Dirichlet polynomials and applications to linear equations with prime variables
cites work: A large sieve density estimate near \(\sigma = 1\)
cites work: On the Waring–Goldbach Problem: Exceptional Sets for Sums of Cubes and Higher Powers
cites work: On generalized quadratic equations in three prime variables
cites work: Sums of almost equal prime squares
cites work: On sums of a prime and four prime squares in short intervals
cites work: Hua's theorem on five almost equal prime squares
cites work: Hua's theorem with five amost equal prime variables
cites work: Q3590837
cites work: On exponential sums over primes in short intervals
cites work: Hua's theorem with nine almost equal prime variables
cites work: Q3152747
cites work: Sums of four cubes of primes.
cites work: Q3152226
cites work: The Hardy-Littlewood method.
DOI: 10.1007/S10474-009-9061-6
description: scientific article; zbMATH DE number 5848590