Goldbach’s problem with primes in arithmetic progressions and in short intervals
Journal de théorie des nombres de Bordeaux, Tome 25 (2013) no. 2, pp. 331-351.

Nous discutons quelques théorèmes sur des valeurs moyennes dans le style du théorème de Bombieri-Vinogradov. Ils concernent des problèmes additifs binaires et ternaires avec des nombres premiers dans des progressions arithmétiques et des intervals courts. Nous donnons des estimations non-triviales pour certaines de ces valeurs moyennes. Comme application entre autres, nous démontrons que pour n¬1(6) grand et impair, le problème n=p 1 +p 2 +p 3 de Goldbach a une solution avec des nombres premiers p 1 ,p 2 dans des intervals courts : p i [X i ,X i +Y], où X i θ i =Y et θ i 0.933 pour i=1,2, et tel que en plus, (p 1 +2)(p 2 +2) a au plus 9 facteurs premiers.

Some mean value theorems in the style of Bombieri-Vinogradov’s theorem are discussed. They concern binary and ternary additive problems with primes in arithmetic progressions and short intervals. Nontrivial estimates for some of these mean values are given. As application inter alia, we show that for large odd n¬1(6), Goldbach’s ternary problem n=p 1 +p 2 +p 3 is solvable with primes p 1 ,p 2 in short intervals p i [X i ,X i +Y] with X i θ i =Y, i=1,2, and θ 1 ,θ 2 0.933 such that (p 1 +2)(p 2 +2) has at most 9 prime factors.

DOI : 10.5802/jtnb.839
Mots clés : additive problems, circle method, sieve methods, short intervals.
Halupczok, Karin 1

1 Universität Münster Mathematisches Institut Einsteinstr. 62 D-48149 Münster, Germany
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Halupczok, Karin. Goldbach’s problem with primes in arithmetic progressions and in short intervals. Journal de théorie des nombres de Bordeaux, Tome 25 (2013) no. 2, pp. 331-351. doi : 10.5802/jtnb.839. http://archive.numdam.org/articles/10.5802/jtnb.839/

[1] A. Balog, A.C. Cojocaru and C. David, Average twin prime conjecture for elliptic curves. Amer. J. Math. 133 (2011), no. 5, 1179–1229. | MR

[2] J. Brüdern, Einführung in die analytische Zahlentheorie. Springer-Lehrbuch, 1995. | Zbl

[3] H. Halberstam and H.-E. Richert, Sieve methods. London Mathematical Society Monographs, No. 4. Academic Press, London-New York, 1974. | MR | Zbl

[4] K. Halupczok, On the ternary Goldbach problem with primes in independent arithmetic progressions. Acta Math. Hungar. 120 (4) (2008), 315–349. | MR | Zbl

[5] K. Kawada, The prime k-tuplets in arithmetic progressions. Tsukuba J. Math. 17, No. 1 (1993), 43–57. | MR | Zbl

[6] M. B. S. Laporta, A short intervals result for 2n-twin primes in arithmetic progressions. Tsukuba J. Math. 23 (1999), no. 2, 201–214. | MR | Zbl

[7] X. Meng, A mean value theorem on the binary Goldbach problem and its application. Monatsh. Math. 151 (2007), 319–332. | MR | Zbl

[8] X. Meng, On linear equations with prime variables of special type. Journal of Number Theory 129 (2009), 2504–2518. | MR | Zbl

[9] H. Mikawa, On prime twins in arithmetic progressions. Tsukuba J. Math. vol. 16 no. 2 (1992), 377–387. | MR | Zbl

[10] H. Mikawa, On prime twins. Tsukuba J. Math. 15 (1991), 19–29. | MR | Zbl

[11] A. Perelli and J. Pintz, On the exceptional set for Goldbach’s problem in short intervals. J. London Math. Soc. (2) 47 (1993), no. 1, 41–49. | MR | Zbl

[12] A. Perelli, J. Pintz and S. Salerno, Bombieri’s theorem in short intervals. Ann. Scuola Normale Sup. Pisa 11 (1984), 529–539. | Numdam | MR | Zbl

[13] B. Saffari and R. C. Vaughan, On the fractional parts of x/n and related sequences II. Ann. Inst. Fourier 27 (1977), 1–30. | Numdam | MR | Zbl

[14] D. I. Tolev, On the number of representations of an odd integer as a sum of three primes, one of which belongs to an arithmetic progression. Tr. Mat. Inst. Steklova 218 (1997), Anal. Teor. Chisel i Prilozh., 415–432; translation in Proc. Steklov Inst. Math. 1997, no. 3 (218), 414–432. | MR | Zbl

[15] D. I. Tolev, The ternary Goldbach problem with arithmetic weights attached to two of the variables. Journal of Number Theory 130 (2010) 439–457. | MR | Zbl

[16] D. I. Tolev, The ternary Goldbach problem with primes from arithmetic progressions. Q. J. Math. 62 (2011), no. 1, 215–221. | MR | Zbl

[17] J. Wu, Théorèmes généralisés de Bombieri-Vinogradov dans les petits intervalles. Quart. J. Math. Oxford Ser. (2) 44 (1993), no. 173, 109–128. | MR | Zbl

[18] J. Wu, Chen’s double sieve, Goldbach’s conjecture and the twin prime problem. Acta Arith. 114 (2004), no. 3, 215–273. | MR | Zbl

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