On Goldbach's conjecture in arithmetic progressions (Q2754989)

This paper presents a generalization of the famous theorem of Vinogradov that says: every sufficiently large positive odd integer is representable as the sum of three primes. Here the author proves the following result: NEWLINENEWLINENEWLINELet \(N\) be any odd positive integer, and \(r, b_1,b_2,b_3\) positive integers with \((r,b_i)=1\). If \(N\equiv b_1 + b_2 + b_3 \pmod r\), then there exists a computable constant \(\delta\) such that for any sufficiently large N and any \(r\leq N^{\delta}\), the equation \(N=p_1+p_2+p_3\) has prime solutions \(p_1,p_2,p_3\), which satisfy \(p\equiv b_i\pmod r\). NEWLINENEWLINENEWLINEThe methods used in the proof are based on a modification, established by \textit{M.-C. Liu} and \textit{K.-M. Tsang} [Théorie des nombres (Quebec, 1987), 595-624 (1987; Zbl 0682.10043)], of the techniques of \textit{H. L. Montgomery} and \textit{R. C. Vaughan} [Acta Arth. 27, 353-370 (1975; Zbl 0301.10043)].