On estimations of trigonometric sums over primes in short intervals. II
type: Item
instance of: scholarly article
title: On estimations of trigonometric sums over primes in short intervals. II
zbMATH Open document ID: 0684.10034
author: Chengdong Pan
author: Chengbiao Pan
published in: Science in China. Series A
publication date: 1989-01-01T00:00:00Z
review text: [For part I, see Sci. China, Ser. A 32, No.4, 408-416 (1989; Zbl 0671.10032).] The ternary Goldbach problem with almost equal primes is concerned with the solutions to \(N=p_ 1+p_ 2+p_ 3\) where N is a given large odd number and \(p_ i\) are primes satisfying \(| p_ i-p_ j| \ll N^{\theta}\) for some \(\theta <1\). The circle method of solution requires estimates for \[ S(\alpha;x,A)=\sum_{x-A<n\leq x}\Lambda (n)e(n\alpha), \] where \(\Lambda\) (n) is the von Mangoldt function and \(e(t)=\exp (2\pi it)\). The authors use an analytic method to prove that, if \(91/96<\theta <1\) and \(c>0\) then there are positive \(c_ 1\), \(c_ 2\) such that \[ S(a/q+\lambda;x,A)\quad \ll \quad A(\log x)^{-c}\quad for\quad x^{\theta}\leq A\leq x, \] provided that \((a,q)=1\), \(1\leq q\leq \log^{c_ 1}x\) and \(\log^{c_ 2}x/A<| \lambda | \leq 1/(q \log^{c_ 1}x)\). For such \(\theta\) and large odd N, the number of solutions to the ternary Goldbach problem is given by the usual asymptotic formula. The authors also point out that their proof here fills some gaps in the earlier papers by the first author [Acta Math. Sin. 9, 315-329 (1959; Zbl 0148.023)] and by \textit{J. Chen} [Sci. Sin. 14, 1113-1117 (1965; Zbl 0146.273)].
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 11P55
Mathematics Subject Classification ID: 11P32
zbMATH DE Number: 4120264
zbMATH Keywords: sums over primes in short intervals
zbMATH Keywords: exponential sums
zbMATH Keywords: ternary Goldbach problem
zbMATH Keywords: circle method
zbMATH Keywords: von Mangoldt function
zbMATH Keywords: asymptotic formula
reviewed by: Peter Shiu
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4120264