On estimations of trigonometric sums over primes in short intervals. II
[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)].
scientific article; zbMATH DE number 4120264