# The exceptional set of Goldbach numbers (II)

Acta Arithmetica (2000)

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

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topHongze 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

top- [1] J. R. Chen, The exceptional set of Goldbach numbers (II), Sci. Sinica 26 (1983), 714-731. Zbl0513.10045
- [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. Zbl0439.10034
- [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] H. Z. Li, Zero-free regions for Dirichlet L-functions, Quart. J. Math. Oxford Ser. (2) 50 (1999), 13-23. Zbl0934.11042
- [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. Zbl0924.11086
- [8] H. L. Montgomery and R. C. Vaughan, The exceptional set in Goldbach's problem, Acta Arith. 27 (1975), 353-370. Zbl0301.10043
- [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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