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On prime twins. (English) Zbl 0735.11041

Let \(\Psi(y,2k)=\sum_{2k<n\leq y}\Lambda(n)\Lambda(n-2k)\) denote the weighted prime twin function with difference \(2k\). It was shown by the reviewer [Math. Ann. 283, 529-537 (1989; Zbl 0646.10033)] that the Hardy- Littlewood formula \(\Psi(y,2k)\sim\sigma(2k)(y-2k)\) \((y\to\infty)\) is true in the range \(2x\leq y\leq x^{8/5-\epsilon}\) for all numbers \(k\leq x\) with at most \(O(x(\log(x))^{-A})\) exceptions. The exponent \(8/5- \epsilon\) was improved to \(2-\epsilon\) by A. Perelli and J. Pintz (to appear in Compos. Math.)
The present author attains the larger interval \(2x\leq y\leq x^{3- \epsilon}\). The proof rests on a deep bound for the mean value of a short exponential sum over primes.

MSC:

11N05 Distribution of primes
11P55 Applications of the Hardy-Littlewood method
11L20 Sums over primes
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