The ternary Goldbach problem in arithmetic progressions

Jianya Liu; Tao Zhan

Jianya Liu ; Tao Zhan

Acta Arithmetica, Tome 80 (1997), p. 197-227 / Harvested from The Polish Digital Mathematics Library

Résumé

For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and $(N, r) = b \in ℕ ³ : 1 \leq b_{j} \leq r, (b_{j}, r) = 1 a n d b ₁ + b ₂ + b ₃ \equiv N (m o d r) .$ It is known that $(N, r) = r ² \prod_{p | r} p | N ((p - 1) (p - 2) / p ²) \prod_{p | r} p ∤ N ((p ² - 3 p + 3) / p ²)$ . Let ε > 0 be arbitrary and $R = N^{1 / 8 - ε}$ . We prove that for all positive integers r ≤ R, with at most $O (R l o g^{- A} N)$ exceptions, the Diophantine equation ⎧N = p₁+p₂+p₃, ⎨ $p_{j} \equiv b_{j} (m o d r),$ j = 1,2,3,⎩ with prime variables is solvable whenever b ∈ (N,r), where A > 0 is arbitrary.

Publié le : 1997-01-01

Zbl 0889.11035

EUDML-ID : urn:eudml:doc:207088

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     title = {The ternary Goldbach problem in arithmetic progressions},
     journal = {Acta Arithmetica},
     volume = {80},
     year = {1997},
     pages = {197-227},
     zbl = {0889.11035},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav82i3p197bwm}
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Jianya Liu; Tao Zhan. The ternary Goldbach problem in arithmetic progressions. Acta Arithmetica, Tome 80 (1997) pp. 197-227. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav82i3p197bwm/

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