# The ternary Goldbach problem in arithmetic progressions

Acta Arithmetica (1997)

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

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topJianya 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 -

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## Citations in EuDML Documents

top- Karin Halupczok, On the ternary Goldbach problem with primes in arithmetic progressions having a common modulus
- Yonghui Wang, Numbers representable by five prime squares with primes in an arithmetic progression
- Maurizio Laporta, On the number of representations in the Waring-Goldbach problem with a prime variable in an arithmetic progression

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