## 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

### Keywords:

additive problems; circle method; sieve methods; short interval
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### References:

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