The exceptional set of Goldbach numbers (II)
Hongze Li
Acta Arithmetica, Tome 92 (2000), p. 71-88 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

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(x1-Δ) for some positive constant Δ > 0.In[3]ChenandPanprovedthatonecantakeΔ>0.01.In[6],weprovedthatE(x)=O(x0.921). In this paper we prove the following result. Theorem. For sufficiently large x, E(x)=O(x0.914). 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=Y1+ε.

Publié le : 2000-01-01
EUDML-ID : urn:eudml:doc:207370 
@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