Sums of five almost equal prime squares. II
type: Item
instance of: scholarly article
title: Sums of five almost equal prime squares. II
zbMATH Open document ID: 0938.11048
DOI: 10.1007/BF02901953
published in: Science in China. Series A
publication date: 1998-10-06T00:00:00Z
review text: [Part I, cf. Acta Arith. 77, 369-383 (1996; Zbl 0863.11067).] The main result obtained in this paper is that every integer \(N\equiv 5\pmod {24}\) can be represented as the sum of five squares of primes of almost equal magnitude. More precisely, the authors show that the five primes can all be chosen from the integral \([\sqrt{N/5}-U, \sqrt{N/5}+U]\) where \(U+ N^{12/25+ \varepsilon}\) for any \(\varepsilon> 0\). The key feature of this result lies in that \(U\) is smaller than \(\sqrt{N}\) by a power of \(N\). This is achieved by choosing the minor arcs considerably smaller so that for \(\alpha\) in the minor arcs, \[ \sum_{|n- \sqrt{N/5} |\leq U} \Lambda(n) e(n^2 \alpha)\ll U^2 N^{-\frac 12- \delta} \] for some \(\delta> 0\). Naturally, the widened major arcs are then more difficult to handle. For this, the authors follow the argument used in the paper [Monatsh. Math. 111, 147-169 (1991; Zbl 0719.11064)] of \textit{M. C. Liu} and the reviewer. Since the authors do not aim at a sharp estimate of the number of representations of \(N\), the influence of the possible Siegel's zero is avoided.
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11L20
zbMATH DE Number: 1207229
zbMATH Keywords: Goldbach-type theorem
zbMATH Keywords: exponential sum over primes
author: Jianya Liu
author: Tao Zhan
reviewed by: Kai-Man Tsang
MaRDI profile type: Publication
cites work: Q5587215
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full work available at URL: https://doi.org/10.1007/bf02901953
OpenAlex ID: W2052564515
description: scientific article; zbMATH DE number 1207229