A short intervals result for \(2n\)-twin primes in arithmetic progressions. (English) Zbl 0941.11032

Let \(I_{k,l}(2n)=\sum\log p_1\log p_2\), where the summation is taken over the pairs of primes \(p_1\) and \(p_2\) such that \(p_1-p_2=2n\), \(p_1\equiv l\) (mod \(k\)) and \(p_2<2n\). When \((l,k)=(2n-l,k)=1\), it is expected that \(I_{k,l}(2n)\sim 2nS_{k,l}(2n)/\varphi(k)\), as \(n\rightarrow\infty\), where \(\varphi(k)\) denotes Euler’s totient function, and \[ S_{k,l}(2n)=2\prod_{p>2}\bigl(1-(p-1)^{-2}\bigr) \prod_{p|nk, p>2}\bigl((p-1)/(p-2)\bigr). \] Then the auther proves the following; Let \(A>0\) and \(0<\varepsilon<2/3\) be arbitrary constants, assume that \(N^{1/3+\varepsilon}\leq H\leq N\), and set \(S_{k,l}(2n)=0\) when \((2n-l,k)>1\). Then there exists a constant \(B=B(A)\) such that \[ \sum_{k\leq K(N)}\max_{(l,k)=1}\sum_{N<2n\leq N+H} \bigl|I_{k,l}(2n)-2nS_{k,l}(2n)/\varphi(k)\bigr|\ll HN(\log N)^{-A}, \] where \(K(N)=\min\{H(\log N)^{-2A-6},\sqrt{N}(\log N)^{-B}\}\), and the implied constant depends only on \(A\) and \(\varepsilon\).
The author also considers a sum which is defined like \(I_{k,l}(2n)\), but the size of primes \(p_2\) is restricted to be smaller, roughly speaking. He establishes a similar result for the latter sum, which contains the above conclusion. The proof is based on the Hardy-Littlewood method, and requires various ideas which appeared in the research for this kind of problems.


11N13 Primes in congruence classes
11P32 Goldbach-type theorems; other additive questions involving primes
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