Abstract
We show that for every fixed A > 0 and θ > 0 there is a ϑ = ϑ(A, θ) > 0 with the following property. Let n be odd and sufficiently large, and let Q 1 = Q 2:= n 1/2(log n)−ϑ and Q 3:= (log n)θ. Then for all q 3 ≦ Q 3, all reduced residues a 3 mod q 3, almost all q 2 ≦ Q 2, all admissible residues a 2 mod q 2, almost all q 1 ≦ Q 1 and all admissible residues a 1 mod q 1, there exists a representation n = p 1 + p 2 + p 3 with primes p i ≡ a i (q i ), i = 1, 2, 3.
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Halupczok, K. On the ternary goldbach problem with primes in independent arithmetic progressions. Acta Math Hung 120, 315–349 (2008). https://doi.org/10.1007/s10474-008-7068-z
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DOI: https://doi.org/10.1007/s10474-008-7068-z
Key words and phrases
- ternary Goldbach problem with primes in residue classes
- Hardy-Littlewood circle method
- applications of the large sieve