Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-05T14:22:20.070Z Has data issue: false hasContentIssue false
  • English
  • Français

Linnik's theorem on Goldbach numbers in short intervals

Published online by Cambridge University Press:  18 May 2009

D. A. Goldston
D. A. Goldston
Affiliation:
Department of Mathematics and Computer ScienceSan Jose State UniversitySan Jose, CA 95192USA
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The Goldbach conjecture states that every even number larger than 2 can be written as the sum of two primes. We shall therefore call an even number a Goldbach number if it can be written as the sum of two primes in at least one way. It has been known for a long time that almost all even numbers are Goldbach numbers. In fact, Montgomery and Vaughan [14] have shown that if E(N) denotes the number of even numbers less than or equal to N which are not Goldbach numbers, then there exists an absolute constant δ>0 such that

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 32 , Issue 3 , September 1990 , pp. 285 - 297
Copyright
Copyright © Glasgow Mathematical Journal Trust 1990

References

REFERENCES

1.Chen, J. R., On the Goldbach's problem and the sieve methods, Sci. Sinica 21 (1978), 701–739.Google Scholar
2.Fouvry, E. and Grupp, F., On the switching principle in sieve theory, J. Reine Angew. Math. 370 (1986), 101–126.Google Scholar
3.Gallagher, P. X., A large sieve density estimate near σ = 1, Invent. Math. 11 (1970), 329–339.Google Scholar
4.Goldston, D. A., The second moment for prime numbers, Quart. J. Math. Oxford (2) 35 (1984), 153–163.CrossRefGoogle Scholar
5.Goldston, D. A. and Montgomery, H. L., Pair correlation of zeros and primes in short intervals, Analytic Number Theory and Diophantine Problems, (Birkhaüser 1987), 183–203.CrossRefGoogle Scholar
6.Halberstam, H. and Richert, H.-E., Sieve Methods (Academic Press, 1974).Google Scholar
7.Hardy, G. H. and Littlewood, J. E., Some Problems of ‘Partitio Numerorum’: III, On the expression of a number as a sum of primes, Acta. Math. 44 (1923), 1–70.CrossRefGoogle Scholar
8.Heath-Brown, D. R., Primes in ‘almost all’ short intervals, J. London Math. Soc. (2) 26 (1982), 385–396.CrossRefGoogle Scholar
9.Keng, Hua Loo, Introduction to Number Theory (Springer-Verlag, 1982).CrossRefGoogle Scholar
10.Ivić, Aleksandar, The Riemann Zeta-Function (John Wiley and Sons, 1985).Google Scholar
11.Kátai, I., A remark on a paper of Ju. V. Linnik, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 17 (1967), 99–100.Google Scholar
12.Linnik, Yu. V., Some conditional theorems concerning the binary Goldbach problem, Izv. Akad. Nauk SSSR 16 (1952), 503–520.Google Scholar
13.Montgomery, H. L., Topics in Multiplicative Number Theory, Lecture Notes in Mathematics 227 (Springer-Verlag, 1971).Google Scholar
14.Montgomery, H. L. and Vaughan, R. C., The exceptional set in Goldbach's problem, Acta Arith. 27 (1975), 353–370.Google Scholar
15.Ramachandra, K., On the number of Goldbach numbers in small intervals, J. Indian Math. Soc. 37 (1973), 157–170.Google Scholar
16.Ramachandra, K., Two remarks in prime number theory, Bull. Soc. Math. France 105 (1977), 433–437.CrossRefGoogle Scholar
17.Selberg, A., On the normal density of primes in small intervals, and the difference between consecutive primes, Arch. Math. Naturvid. 47 (1943), no. 6, 87–105.Google Scholar