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

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.

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.

DOI : https://doi.org/10.5802/jtnb.839
Keywords: additive problems; circle method; sieve methods; short intervals.
@article{JTNB_2013__25_2_331_0,
     author = {Halupczok, Karin},
     title = {Goldbach's problem with primes in arithmetic progressions and in short intervals},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {25},
     number = {2},
     year = {2013},
     pages = {331-351},
     doi = {10.5802/jtnb.839},
     mrnumber = {3228311},
     zbl = {1294.11173},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2013__25_2_331_0}
}
Halupczok, Karin. Goldbach’s problem with primes in arithmetic progressions and in short intervals. Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 2, pp. 331-351. doi : 10.5802/jtnb.839. http://www.numdam.org/item/JTNB_2013__25_2_331_0/

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