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1997 | 82 | 3 | 197-227
Tytuł artykułu

The ternary Goldbach problem in arithmetic progressions

Autorzy
Treść / Zawartość
Warianty tytułu
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.
Czasopismo
Rocznik
Tom
82
Numer
3
Strony
197-227
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-06-08
poprawiono
1996-12-06
Twórcy
autor
autor
  • Department of Mathematics, Shandong University, Jinan, Shandong 250100, P.R. China
Bibliografia
  • [A] Ayoub R., On Rademacher's extension of the Goldbach-Vinogradov theorem, Trans. Amer. Math. Soc. 74 (1953), 482-491.
  • [B] Balog A., The prime k-tuplets conjecture on average, in: Analytic Number Theory (Allerton Park, Ill.), Birkhäuser, 1990, 47-75.
  • [BP] Balog A. and Perelli A., Exponential sums over primes in an arithmetic progression, Proc. Amer. Math. Soc. 93 (1985), 578-582.
  • [D] Davenport H., Multiplicative Number Theory, revised by H. L. Montgomery, Springer, 1980.
  • [HL] Hardy G. H. and Littlewood J. E., Some problems of 'partitio numerorumi' III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1-70.
  • [La] Lavrik A. F., The number of k-twin primes lying on an interval of a given length, Dokl. Akad. Nauk SSSR 136 (1961), 281-283 (in Russian); English transl.: Soviet Math. Dokl. 2 (1961), 52-55.
  • [Li] Liu J. Y., The Goldbach-Vinogradov theorem with primes in a thin subset, Chinese Ann. Math., to appear.
  • [LT] Liu M. C. and Tsang K. M., Small prime solutions of linear equations, in: Théorie des Nombres, J.-M. De Koninck and C. Levesque (eds.), de Gruyter, Berlin, 1989, 595-624.
  • [MP] Maier H. and Pomerance C., Unusually large gaps between consecutive primes, Trans. Amer. Math. Soc. 332 (1990), 201-237.
  • [Mi] Mikawa H., On prime twins in arithmetic progressions, Tsukuba J. Math. 16 (1992), 377-387.
  • [Mo] Montgomery H. L., Topics in Multiplicative Number Theory, Lecture Notes in Math. 227, Springer, 1971.
  • [R] Rademacher H., Über eine Erweiterung des Goldbachschen Problems, Math. Z. 25 (1926), 627-660.
  • [T] Titchmarsh E. C., The Theory of the Riemann Zeta-Function, 2nd ed., revised by D. R. Heath-Brown, Oxford Univ. Press, 1986.
  • [Va] Vaughan R. C., An elementary method in prime number theory, in: Recent Progress in Analytic Number Theory, H. Halberstam and C. Hooley (eds.), Vol. 1, Academic Press, 1981, 341-348.
  • [Vi] Vinogradov I. M., Some theorems concerning the theory of primes, Mat. Sb. (N.S.) 2 (1937), 179-195.
  • [W] Wolke D., Some applications to zero-density theorems for L-functions, Acta Math. Hungar. 61 (1993), 241-258.
  • [Zh] Zhan T., On the representation of odd number as sums of three almost equal primes, Acta Math. Sinica (N.S.) 7 (1991), 259-275.
  • [ZL] Zhan T. and Liu J. Y., A Bombieri-type mean-value theorem concerning exponential sums over primes, Chinese Sci. Bull. 41 (1996), 363-366.
  • [Zu] Zulauf A., Beweis einer Erweiterung des Satzes von Goldbach-Vinogradov, J. Reine Angew. Math. 190 (1952), 169-198.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-aav82i3p197bwm
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