The exceptional set of Goldbach numbers (II)

Hongze Li

Acta Arithmetica (2000)

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

Abstract

top
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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.