The ternary Goldbach problem with primes in arithmetic progressions (Q5958938)

The authors prove the following arithmetic-progression version of the Goldbach-Vinogradov theorem. Let \(N\) be odd and sufficiently large, \(1\leq k\leq N^{1/42}\), \((\ell_j,k)=1\) \((1\leq j\leq 3)\), \(N\equiv \ell_1+ \ell_2+ \ell_3\bmod k\). Then the equation \(N= p_1+ p_2+ p_3\) with \(p_j\equiv \ell_j\bmod k\) \((1\leq j\leq 3)\) is solvable in primes \(p_1, p_2, p_3\). This improves a result of \textit{M.-C. Liu} and \textit{T. Zhan} [Motohashi, Y. (ed.), Analytic number theory, Kyoto 1996, Lond. Math. Soc. Lect. Note Ser. 247, 227-251 (1997; Zbl 0913.11043)]. The proof follows the classical lines. The Hardy-Littlewood method, and zero density results for the \(L\)-functions are used.