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On the exceptional set for the 2\(k\)-twin primes problem. (English) Zbl 0756.11028
The classical Hardy-Littlewood conjecture states, in a slightly modified form, \[ \psi(N,2k)={\mathfrak S}(2k)(N-2k)+O(N(\ln N)^{-A}),\tag{*} \] where \(k\) is a fixed positive integer, \[ \psi(N,2k)=\sum_{\textstyle{{m,n=N}\atop{n- m=2k}}}^{2N}\Lambda(m)\Lambda(n), \qquad {\mathfrak S}(2k)=2\prod_{p>2}\left(1-{1\over(p-1)^ 2}\right)\prod_{\textstyle{{p\mid k}\atop {p>2}}}{p-1 \over p-2}, \] and \(A\) is any positive number. Several “almost all” results are well known [H. L. Montgomery, Topics in multiplicative number theory (Lect. Notes Math. 227) (Springer 1971; Zbl 0216.03501); D. Wolke, Math. Ann. 283, 529-537 (1989; Zbl 0646.10033)].
By a very careful use of the circle method and zero density results for \(L\)-functions the authors prove the following strong statement. Let \(A,B>0\), \(0\leq V<N/4\), and \(N^{1/2+\varepsilon}<N<N/4\) (\(0<\varepsilon<1/2\)). Then (*) holds for all \(V\leq k\leq V+H\) but \(O(H(\ln N)^{-B})\) exceptions.
They indicate that a similar result is true for the binary Goldbach problem, i.e. the Hardy-Littlewood formula for the number of binary Goldbach representations is valid for all even numbers \(2n\in I\) but \(O(| I|(\ln x)^{-B})\) exceptions, where \(I\) is the interval \([x,x+x^{1/2+\varepsilon}]\).

MSC:
11P32 Goldbach-type theorems; other additive questions involving primes
11N05 Distribution of primes
11P55 Applications of the Hardy-Littlewood method
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