On the ternary Goldbach problem with primes in arithmetic progressions having a common modulus
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, p. 203-213

For ε>0 and any sufficiently large odd n we show that for almost all kR:=n 1/5-ε there exists a representation n=p 1 +p 2 +p 3 with primes p i b i mod k for almost all admissible triplets b 1 ,b 2 ,b 3 of reduced residues mod k.

Pour ε>0 et n impair suffisamment grand, nous montrons que, pour presque tout kR:=n 1/5-ε , il existe une représentation n=p 1 +p 2 +p 3 avec des nombres premiers p i b i modulo k pour presque tout triplet admissible b 1 ,b 2 ,b 3 de résidus modulo k.

@article{JTNB_2009__21_1_203_0,
     author = {Halupczok, Karin},
     title = {On the ternary Goldbach problem with primes in arithmetic progressions having a common modulus},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {1},
     year = {2009},
     pages = {203-213},
     doi = {10.5802/jtnb.666},
     mrnumber = {2537712},
     zbl = {pre05620677},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2009__21_1_203_0}
}
Halupczok, Karin. On the ternary Goldbach problem with primes in arithmetic progressions having a common modulus. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, pp. 203-213. doi : 10.5802/jtnb.666. http://www.numdam.org/item/JTNB_2009__21_1_203_0/

[1] A. Balog, The Prime k-Tuplets Conjecture on Average. Analytic number theory, Proc. Conf. in Honor of Paul T. Bateman, Urbana/IL (USA), 1989, Prog. Math. 85 (1990), 47–75. | MR 1084173 | Zbl 0719.11066

[2] C. Bauer, Y. Wang, On the Goldbach conjecture in arithmetic progressions. Rocky Mountain J. Math. 36 (1) (2006), 35–66. | MR 2228183 | Zbl 1148.11053

[3] Z. Cui, The ternary Goldbach problem in arithmetic progression II. Acta Math. Sinica (Chin. Ser.) 49 (1) (2006), 129–138. | MR 2248920 | Zbl pre05186362

[4] J. Liu, T. Zhang, The ternary Goldbach problem in arithmetic progressions. Acta Arith. 82 (3) (1997), 197–227. | MR 1482887 | Zbl 0889.11035

[5] M. B. Nathanson, Additive Number Theory: The Classical Bases. Graduate texts in Mathematics 164, Springer-Verlag, 1996. | MR 1395371 | Zbl 0859.11002