The exceptional set of Goldbach numbers (II)

Hongze Li

Acta Arithmetica (2000)

  • Volume: 92, Issue: 1, page 71-88
  • ISSN: 0065-1036

Abstract

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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 . I n [ 3 ] C h e n a n d P a n p r o v e d t h a t o n e c a n t a k e Δ > 0 . 01 . I n [ 6 ] , w e p r o v e d t h a t E ( 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 .

How to cite

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Hongze Li. "The exceptional set of Goldbach numbers (II)." Acta Arithmetica 92.1 (2000): 71-88. <http://eudml.org/doc/207370>.

@article{HongzeLi2000,
abstract = {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 $D = Y^\{1+ε\}$.},
author = {Hongze Li},
journal = {Acta Arithmetica},
keywords = {exceptional set; Goldbach conjecture},
language = {eng},
number = {1},
pages = {71-88},
title = {The exceptional set of Goldbach numbers (II)},
url = {http://eudml.org/doc/207370},
volume = {92},
year = {2000},
}

TY - JOUR
AU - Hongze Li
TI - The exceptional set of Goldbach numbers (II)
JO - Acta Arithmetica
PY - 2000
VL - 92
IS - 1
SP - 71
EP - 88
AB - 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 $D = Y^{1+ε}$.
LA - eng
KW - exceptional set; Goldbach conjecture
UR - http://eudml.org/doc/207370
ER -

References

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  1. [1] J. R. Chen, The exceptional set of Goldbach numbers (II), Sci. Sinica 26 (1983), 714-731. Zbl0513.10045
  2. [2] J. R. Chen and J. M. Liu, The exceptional set of Goldbach numbers (III), Chinese Quart. J. Math. 4 (1989), 1-15. 
  3. [3] J. R. Chen and C. D. Pan, The exceptional set of Goldbach numbers, Sci. Sinica 23 (1980), 416-430. Zbl0439.10034
  4. [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. Zbl0739.11033
  5. [5] H. Z. Li, Zero-free regions for Dirichlet L-functions, Quart. J. Math. Oxford Ser. (2) 50 (1999), 13-23. Zbl0934.11042
  6. [6] H. Z. Li, The exceptional set of Goldbach numbers, ibid. 50 (1999). 
  7. [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. Zbl0924.11086
  8. [8] H. L. Montgomery and R. C. Vaughan, The exceptional set in Goldbach's problem, Acta Arith. 27 (1975), 353-370. Zbl0301.10043
  9. [9] W. Wang, On zero distribution of Dirichlet's L-functions, J. Shandong Univ. 21 (1986), 1-13 (in Chinese). Zbl0615.10050

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