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 for some positive constant Δ > 0. In this paper we prove the following result. Theorem. For sufficiently large x, . 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 .
@article{bwmeta1.element.bwnjournal-article-aav92i1p71bwm, author = {Hongze Li}, title = {The exceptional set of Goldbach numbers (II)}, journal = {Acta Arithmetica}, volume = {92}, year = {2000}, pages = {71-88}, zbl = {0963.11057}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav92i1p71bwm} }
Hongze Li. The exceptional set of Goldbach numbers (II). Acta Arithmetica, Tome 92 (2000) pp. 71-88. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav92i1p71bwm/
[00000] [1] J. R. Chen, The exceptional set of Goldbach numbers (II), Sci. Sinica 26 (1983), 714-731. | Zbl 0513.10045
[00001] [2] J. R. Chen and J. M. Liu, The exceptional set of Goldbach numbers (III), Chinese Quart. J. Math. 4 (1989), 1-15.
[00002] [3] J. R. Chen and C. D. Pan, The exceptional set of Goldbach numbers, Sci. Sinica 23 (1980), 416-430. | Zbl 0439.10034
[00003] [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. | Zbl 0739.11033
[00004] [5] H. Z. Li, Zero-free regions for Dirichlet L-functions, Quart. J. Math. Oxford Ser. (2) 50 (1999), 13-23. | Zbl 0934.11042
[00005] [6] H. Z. Li, The exceptional set of Goldbach numbers, ibid. 50 (1999).
[00006] [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. | Zbl 0924.11086
[00007] [8] H. L. Montgomery and R. C. Vaughan, The exceptional set in Goldbach's problem, Acta Arith. 27 (1975), 353-370. | Zbl 0301.10043
[00008] [9] W. Wang, On zero distribution of Dirichlet's L-functions, J. Shandong Univ. 21 (1986), 1-13 (in Chinese). | Zbl 0615.10050