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On estimations of trigonometric sums over primes in short intervals. III. (English) Zbl 0714.11050

This third paper in the series [parts I and II, cf. Sci. China, Ser. A 32, 408-416, 641-653 (1989; Zbl 0671.10032, Zbl 0684.10034)] is still on the ternary Goldbach problem with almost equal primes, that is 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 \(\theta <1\). By applying the recent result of W. Zhan [Chin. Ann. Math., Ser. A 11, No.1, 121-127 (1990; Zbl 0699.10058)] on zero density estimates of L-functions in short intervals, the authors now improve the earlier upper estimates of \(\theta\) in their second paper from 91/96 to 2/3.
Reviewer: P.Shiu

MSC:

11L20 Sums over primes
11P32 Goldbach-type theorems; other additive questions involving primes
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11L03 Trigonometric and exponential sums (general theory)
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