The ternary Goldbach problem in arithmetic progressions

Jianya Liu; Tao Zhan

Acta Arithmetica (1997)

  • Volume: 82, Issue: 3, page 197-227
  • ISSN: 0065-1036

Abstract

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For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and ( N , r ) = b ∈ ℕ ³ : 1 ≤ b j ≤ r , ( b j , r ) = 1 a n d b ₁ + b ₂ + b ₃ ≡ N ( m o d r ) . It is known that    ( N , r ) = r ² ∏ p | r p | N ( ( p - 1 ) ( p - 2 ) / p ² ) ∏ 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 ≡ b j ( m o d r ) , j = 1,2,3, ⎩ with prime variables is solvable whenever b ∈ (N,r), where A > 0 is arbitrary.

How to cite

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Jianya Liu, and Tao Zhan. "The ternary Goldbach problem in arithmetic progressions." Acta Arithmetica 82.3 (1997): 197-227. <http://eudml.org/doc/207088>.

@article{JianyaLiu1997,
abstract = {For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and $(N,r) = \{b∈ ℕ³ : 1 ≤ b_j ≤ r, (b_j, r) = 1 and b₁+b₂+b₃ ≡ N (mod r)\}. $It is known that    $#(N,r) = r² ∏_\{p|r\}_\{p|N\} ((p-1)(p-2)/p²) ∏_\{p|r\}_\{p∤N\} ((p²-3p+3)/p²)$. Let ε > 0 be arbitrary and $R = N^\{1/8-ε\}$. We prove that for all positive integers r ≤ R, with at most $O(Rlog^\{-A\}N)$ exceptions, the Diophantine equation ⎧N = p₁+p₂+p₃, ⎨ $p_j ≡ b_j (mod r),$ j = 1,2,3,$ $⎩ with prime variables is solvable whenever b ∈ (N,r), where A > 0 is arbitrary.},
author = {Jianya Liu, Tao Zhan},
journal = {Acta Arithmetica},
keywords = {ternary Goldbach problem; exponential sum over primes in arithmetic progressions; mean-value theorem; arithmetic progressions; circle method; major-arcs mean-value estimate; exponential sum over primes},
language = {eng},
number = {3},
pages = {197-227},
title = {The ternary Goldbach problem in arithmetic progressions},
url = {http://eudml.org/doc/207088},
volume = {82},
year = {1997},
}

TY - JOUR
AU - Jianya Liu
AU - Tao Zhan
TI - The ternary Goldbach problem in arithmetic progressions
JO - Acta Arithmetica
PY - 1997
VL - 82
IS - 3
SP - 197
EP - 227
AB - For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and $(N,r) = {b∈ ℕ³ : 1 ≤ b_j ≤ r, (b_j, r) = 1 and b₁+b₂+b₃ ≡ N (mod r)}. $It is known that    $#(N,r) = r² ∏_{p|r}_{p|N} ((p-1)(p-2)/p²) ∏_{p|r}_{p∤N} ((p²-3p+3)/p²)$. Let ε > 0 be arbitrary and $R = N^{1/8-ε}$. We prove that for all positive integers r ≤ R, with at most $O(Rlog^{-A}N)$ exceptions, the Diophantine equation ⎧N = p₁+p₂+p₃, ⎨ $p_j ≡ b_j (mod r),$ j = 1,2,3,$ $⎩ with prime variables is solvable whenever b ∈ (N,r), where A > 0 is arbitrary.
LA - eng
KW - ternary Goldbach problem; exponential sum over primes in arithmetic progressions; mean-value theorem; arithmetic progressions; circle method; major-arcs mean-value estimate; exponential sum over primes
UR - http://eudml.org/doc/207088
ER -

References

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