Goldbach's problem with primes in arithmetic progressions and in short intervals (Q394266)

Some mean value theorems in the style of Bombieri-Vinogradov's theorem are discussed.NEWLINENEWLINEThe author obtains the following result.NEWLINENEWLINETheorem. 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 \(i=1,2\).NEWLINENEWLINE Then for any fixed integers \(a_{1}, a_{2}\) with \(a_{1}\leq n-X_{1}-Y\), we have NEWLINE\[NEWLINE\sum\limits_{q_{1}\leq Q_{1}}\sum\limits_{q_{2}\leq Q_{2}}\Big|\sum_{\substack{ p_{1}+p_{2}+p_{3}=n,\\p_{i}\in[X_{i},X_{i}+Y],\\p_{i}\equiv a_{i}\pmod {q_{i}},\, i=1,2}} \log p_{1}\log p_{2}\log p_{3}-\mathfrak{S}(n,q_{1},a_{1},q_{2},a_{2})Y^{2}\Big|\ll Y^{2}L^{-A}.NEWLINE\]