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On the exceptional set for Goldbach’s problem in short intervals. (English) Zbl 0806.11042

The authors continue their investigations on Goldbach numbers in short intervals [Compos. Math. 82, 355-372 (1992; Zbl 0756.11028)]. Let \(R(2n)= \sum_{h+k= 2n} \Lambda(h) \Lambda(k)\) and \(S(2n)= 2 \prod_{p>2} (1- (p-1)^{-2}) \prod _{\substack{ p\mid n\\ p>2}} {{p-1} \over {p-2}}\). Then, for \(0<\varepsilon< 2/3\), \(A>0\), and \(N^{1/3+\varepsilon}\leq H\leq N\), \[ \sum_{N\leq 2n\leq N+H} | R(2n)- 2n S(2n)|^ 2\ll_{\varepsilon, A} HN^ 2 (\ln N)^{-A}. \] The progress (exponent 1/3 instead of 1/2) is achieved by a more careful estimation of the contribution of the major arcs.

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
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