The ternary Goldbach problem in arithmetic progressions

• Autorzy: Jianya Liu , Tao Zhan
• Języki publikacji: EN

Abstrakty

EN

For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃)
and
$𝓑(N,r) = {b∈ ℕ³ : 1 ≤ b_j ≤ r, (b_j, r) = 1$ and b₁+b₂+b₃ ≡ N (mod r)}.
It is known that
   $#𝓑(N,r) = r² ∏_{p|r}_{p|N} ((p-1)(p-2)/p²) ∏_{p|r}_{p∤N} ((p²-3p+3)/p²)$.

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 ∈ 𝓑(N,r), where A > 0 is arbitrary.

Słowa kluczowe

EN

ternary Goldbach problem; exponential sum over primes in arithmetic progressions; mean-value theorem

Kategorie tematyczne

• 11L07: Estimates on exponential sums
• 11P32: Goldbach-type theorems; other additive questions involving primes

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1997
• Tom: 82
• Numer: 3
• Strony: 197-227

Daty

wydano

1997

otrzymano

1996-06-08

poprawiono

1996-12-06

Twórcy

autor

Jianya Liu

autor

Tao Zhan

• Department of Mathematics, Shandong University, Jinan, Shandong 250100, P.R. China

Bibliografia

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