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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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