Mikawa, Hiroshi On prime twins. (English) Zbl 0735.11041 Tsukuba J. Math. 15, No. 1, 19-29 (1991). 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. Reviewer: D.Wolke (Freiburg i.Br.) Cited in 5 ReviewsCited in 10 Documents MSC: 11N05 Distribution of primes 11P55 Applications of the Hardy-Littlewood method 11L20 Sums over primes Keywords:weighted prime twin function; Hardy-Littlewood formula; mean value; short exponential sum over primes Citations:Zbl 0646.10033; Zbl 0666.10026 PDFBibTeX XMLCite \textit{H. Mikawa}, Tsukuba J. Math. 15, No. 1, 19--29 (1991; Zbl 0735.11041) Full Text: DOI