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Goldbach’s problem with primes in arithmetic progressions and in short intervals. (English. French summary) Zbl 1294.11173

Some mean value theorems in the style of Bombieri-Vinogradov’s theorem are discussed.
The author obtains the following result.
Theorem. Let \(n\geq X_{}+X_{2}+2Y\) be odd, let \(n\ll X_{1}Y\), \(X_{2}\geq Y\gg (n-X_{1})^{2/3+\varepsilon}\), \(X_{1}\geq Y\gg X_{1}^{3/5+\varepsilon}\) and assume that \(Q_{i}\ll YX_{i}^{-1/2}L^{-B}\) for \(i=1,2\).
Then for any fixed integers \(a_{1}, a_{2}\) with \(a_{1}\leq n-X_{1}-Y\), we have \[ \sum\limits_{q_{1}\leq Q_{1}}\sum\limits_{q_{2}\leq Q_{2}}\Big|\sum_{\substack{ p_{1}+p_{2}+p_{3}=n,\\p_{i}\in[X_{i},X_{i}+Y],\\p_{i}\equiv a_{i}\pmod {q_{i}},\, i=1,2}} \log p_{1}\log p_{2}\log p_{3}-\mathfrak{S}(n,q_{1},a_{1},q_{2},a_{2})Y^{2}\Big|\ll Y^{2}L^{-A}. \]

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
11B25 Arithmetic progressions
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References:

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