Hostname: page-component-cb9f654ff-rkzlw Total loading time: 0 Render date: 2025-08-23T10:48:36.924Z Has data issue: false hasContentIssue false
  • English
  • Français

Additive representation in thin sequences, II: The binary Goldbach problem

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  26 February 2010

J. Brüdern ,
K. Kawada  and
T. D. Wooley
J. Brüdern
Affiliation:
Mathematisches Institut A, Universität Stuttgart, Postfach 80 11 40, D-70511 Stuttgart, Germany. E-mail: bruedern@mathematik.uni-stuttgart.de
K. Kawada
Affiliation:
Department of Mathematics, Faculty of Education, Iwate University, Morioka 020-8550, Japan. E-mail: kawada@iwate-u.ac.jp
T. D. Wooley
Affiliation:
Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, MI 48109-1109, U.S.A., E-mail: wooley@math.lsa.umich.edu
Get access

Extract

§1. Introduction. Most prominent among the classical problems in additive number theory are those of Waring and Goldbach type. Although use of the Hardy–Littlewood method has brought admirable progress, the finer questions associated with such problems have yet to find satisfactory solutions. For example, while the ternary Goldbach problem was solved by Vinogradov as early as 1937 (see Vinogradov [16], [17]), the latter's methods permit one to establish merely that almost all even integers are the sum of two primes (see Chudakov [4], van der Corput [5] and Estermann [7]).

MSC classification

Secondary: 11P32: Goldbach-type theorems; other additive questions involving primes

Information

Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 117 - 125
Copyright
Copyright © University College London 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Baker, R. C., Harman, G. and Pintz, J.. The exceptional set for Goldbach's problem in short intervals. Sieve Methods Exponential Sums, and their Applications in Number Theory (Cardiff 1995) (Greaves, G., Harman, G. and Huxley, M., eds.), (Cambridge University Press, Cambridge, 1997), pp. 1–54.Google Scholar
2.Brüdern, J., Kawada, K. and Wooley, T. D.. Additive representation in thin sequences, I: Waring's problem for cubes. Ann. Sci. École Norm. Sup. (4), 34 (2001), 471–501.CrossRefGoogle Scholar
3.Brüdern, J. and Perelli, A.. Goldbach numbers in sparse sequences. Ann. Inst. Fourier, Grenoble, 48 (1998), 353–378.CrossRefGoogle Scholar
4.Chudakov, N. G.. On the density of the set of even numbers which are not representable as a sum of two odd primes. Izv. Akad. Nauk SSSR Ser. Mat., 2 (1938), 25–40.Google Scholar
5.Corput, J. G. van der, Sur l'hypothese de Goldbach. Proc. Akad. Wet. Amsterdam, 41 (1938), 76–80.Google Scholar
6.Davenport, H.. Multiplicative Number Theory (2nd edn.). (Springer-Verlag, Berlin-New York, 1980).CrossRefGoogle Scholar
7.Estermann, T.. On Goldbach's problem: Proof that almost all even positive integers are sums of two primes. Proc. London Math. Soc. (2), 44 (1938), 307–314.CrossRefGoogle Scholar
8.Ford, K. B.. New estimates for mean values of Weyl sums. Internal. Math. Res. Notices (1995), 155–171.CrossRefGoogle Scholar
9.Languasco, A. and Perelli, A.. A pair correlation hypothesis and the exceptional set in Goldbach's problem. Mathematika, 43, (1996), 349–361.CrossRefGoogle Scholar
10.Montgomery, H. L. and Vaughan, R. C.. The exceptional set in Goldbach's problem. Acta Arith., 27 (1975), 353–370.CrossRefGoogle Scholar
11.Perelli, A.. Goldbach numbers represented by polynomials. Rev. Mat. Iberoamericana, 12 (1996), 477–490.CrossRefGoogle Scholar
12.Perelli, A. and Pintz, J.. On the exceptional set for Goldbach's problem in short intervals. J. London Math. Soc. (2), 47 (1993), 41–49.Google Scholar
13.Ramachandra, K.. On the number of Goldbach numbers in small intervals. J. Indian Math. Soc., 37 (1973), 157–170.Google Scholar
14.Vaughan, R. C.. A new iterative method in Waring's problem. Acta Math., 162 (1989), 1–71.CrossRefGoogle Scholar
15.Vaughan, R. C.. The Hardy-Littlewood Method (2nd edn.). (Cambridge University Press, Cambridge, 1997).CrossRefGoogle Scholar
16.Vinogradov, I. M.. Representation of an odd integer as a sum of three primes. C. R. Acad. Sci. URSS, 15 (1937), 6–7.Google Scholar
17.Vinogradov, I. M.. Some theorems concerning the theory of primes. Mat. Sbornik, 2 (1937), 179–195.Google Scholar
18.Wooley, T. D.. Large improvements in Waring's problem. Ann. Math. (2), 135 (1992), 131–164.Google Scholar
19.Wooley, T. D.. The application of a new mean value theorem to the fractional parts of polynomials. Acta Arith., 65 (1993), 163–179.CrossRefGoogle Scholar