## 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.

### MSC:

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