On the representation of large integers as sums of four almost equal squares of primes
type: Item
instance of: scholarly article
title: On the representation of large integers as sums of four almost equal squares of primes
zbMATH Open document ID: 1181.11069
author: Guangshi Lü
author: Wenguang Zhai
published in: The Ramanujan Journal
publication date: 2009-11-23T00:00:00Z
review text: Let \(E(N,U)\) be the number of all positive integers \(n\equiv 4\pmod{24}\) satisfying \(N\leq n\leq N+\sqrt{N}U\), which can not be written as \[ \begin{cases} n=p_1^2+p_2^2+p_3^2+p_4^2;\\ |p_j-\sqrt{\frac{N}{4}}|\leq U,&j=1,2,3,4, \end{cases} \] with \(p_1,p_2,p_3,p_4\) prime numbers. The authors prove: Theorem 1. For \(U=N^{\frac{21}{50}+\varepsilon}\) we have \[ E(N,U)\ll(\sqrt{N}U)^{1-\varepsilon}. \] Theorem 2. For \(U=N^{\frac{9}{20}+\varepsilon}\) we have \[ E(N,U)\ll U^{1-\varepsilon}. \] The proof is based on the new estimates for exponential sums over primes in short intervals and a technique to get the asymptotic formula on the enlarged major arcs in the circle method.
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11P55
Mathematics Subject Classification ID: 11L20
Mathematics Subject Classification ID: 11P05
zbMATH DE Number: 5636933
zbMATH Keywords: sums of four squares of primes
zbMATH Keywords: exponential sums over primes
zbMATH Keywords: circle method
MaRDI profile type: Publication
full work available at URL: s11139-007-9038-4
OpenAlex ID: W2049822316
cites work: On sums of squares of primes
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cites work: On sums of a prime and four prime squares in short intervals
cites work: Q3152226
DOI: 10.1007/S11139-007-9038-4
description: scientific article; zbMATH DE number 5636933