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The exceptional set of Goldbach numbers (II)

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1. Introduction. A positive number which is a sum of two odd primes is called a Goldbach number. Let E(x) denote the number of even numbers not exceeding x which cannot be written as a sum of two odd primes. Then the Goldbach conjecture is equivalent to proving that
E(x) = 2 for every x ≥ 4.
E(x) is usually called the exceptional set of Goldbach numbers. In [8] H. L. Montgomery and R. C. Vaughan proved that $E(x) = O(x^{1-Δ})$ for some positive constant Δ > 0$. In [3] Chen and Pan proved that one can take Δ >0.01. In [6], we proved thatE(x) = O(x^{0.921})$. In this paper we prove the following result.
Theorem. For sufficiently large x,
$E(x) =O (x^{0.914})$.
Throughout this paper, ε always denotes a sufficiently small positive number that may be different at each occurrence. A is assumed to be sufficiently large, A < Y, and $D = Y^{1+ε}$.
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autor
  • Department of Mathematics, Shandong University, Jinan Shandong, P.R. China
Bibliografia
  • [1] J. R. Chen, The exceptional set of Goldbach numbers (II), Sci. Sinica 26 (1983), 714-731.
  • [2] J. R. Chen and J. M. Liu, The exceptional set of Goldbach numbers (III), Chinese Quart. J. Math. 4 (1989), 1-15.
  • [3] J. R. Chen and C. D. Pan, The exceptional set of Goldbach numbers, Sci. Sinica 23 (1980), 416-430.
  • [4] D. R. Heath-Brown, Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. (3) 64 (1992), 265-338.
  • [5] H. Z. Li, Zero-free regions for Dirichlet L-functions, Quart. J. Math. Oxford Ser. (2) 50 (1999), 13-23.
  • [6] H. Z. Li, The exceptional set of Goldbach numbers, ibid. 50 (1999).
  • [7] J. Y. Liu, M. C. Liu and T. Z. Wang, The number of powers of 2 in a representation of large even integers (II), Sci. China Ser. A 41 (1998), 1255-1271.
  • [8] H. L. Montgomery and R. C. Vaughan, The exceptional set in Goldbach's problem, Acta Arith. 27 (1975), 353-370.
  • [9] W. Wang, On zero distribution of Dirichlet's L-functions, J. Shandong Univ. 21 (1986), 1-13 (in Chinese).
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Bibliografia
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bwmeta1.element.bwnjournal-article-aav92i1p71bwm
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