Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-19T12:13:55.866Z Has data issue: false hasContentIssue false
  • English
  • Français

The values of ternary quadratic forms at prime arguments

Published online by Cambridge University Press:  26 February 2010

R. J. Cook  and
A. Fox
R. J. Cook
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Hounsfield RoadSheffield S3 7RHCurrent address: Stoneygate, Bessie Lane, Bradweli, Hope Valley S33 9HZ
A. Fox
Affiliation:
2 Clarence Road, Monk Bretton, Barnsley, South Yorkshire S71 2NL.
Get access

Extract

Montgomery and Vaughan [12] have shown that the exceptional set in Goldbach's problem

satisfies

for some Δ>0. Li [10,11] has shown that we may take Δ = 0·079 and Δ = 0·086. If the Riemann Hypothesis is true for all Dirichlet L-functions then (1) holds for any Δ<½. This is a classical result due to Hardy and Littlewood [7].

Type
Research Article
Information
Mathematika , Volume 48 , Issue 1-2 , December 2001 , pp. 137 - 149
Copyright
Copyright © University College London 2001

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.)

References

1.Bauer, C., Liu, M.-C. and Zhan, T.. On a sum of three prime squares. J. Number Theory 85 (2000). 336–359.CrossRefGoogle Scholar
2.Brudern, J., Cook, R. J. and Perelli, A.. The values of binary linear forms at prime arguments. Sieve Methods, Exponential Sums and their Applications in Number Theory (ed. Greaves, G. R. H., Harman, G. and Huxley, M. N.), (Cambridge University Press, 1996), 87–100.Google Scholar
3.Davenport, H.. Analytic Methods for Diophantine Equations and Diophantine Inequalities. (Campus Publishers, Ann Arbor, Michigan, 1962.).Google Scholar
4.Davenport, H. and Heilbronn, H.. On indefinite quadratic forms in five variables. J. London Math. Soc. 21 (1946), 185–193.CrossRefGoogle Scholar
5.Davenport, H. and Roth, K. F.. The solubility of certain diophantine inequalities. Mathematika 2(1955), 81–96.CrossRefGoogle Scholar
6.Ghosh, A.. The distribution of αp 2 modulo 1. Proc. London Math. Soc. (3) 42 (1981), 252– 269.CrossRefGoogle Scholar
7.Hardy, G. H. and Littlewood, J. E.. Some problems of “Partitio Numerorum”, V. Proc. London Math. Soc. (2) 22 (1923), 46–56.Google Scholar
8.Hua, L. K.. Some results in the additive prime number theory. Quart. J. Math. Oxford 9 (1938), 68–80.CrossRefGoogle Scholar
9.Leung, M.-Ch. and Liu, M.-Ch.. On generalized quadratic equations in three prime variables. Monatsh. Math. 115 (1993), 113–169.CrossRefGoogle Scholar
10.Li, H.The exceptional set of Goldbach numbers. Quart. J. Math Oxford 50 (1999), 471–482.CrossRefGoogle Scholar
11.Li, H.. The exceptional set of Goldbach numbers II. Acta Arith. 92 (2000), 71–88.CrossRefGoogle Scholar
12.Montgomery, H. L. and Vaughan, R. C.. The exceptional set in Goldbach's problem. Acta Arith. 27(1975), 353–370.CrossRefGoogle Scholar
13.Schwarz, W.. Zur Darstellung von Zahlen durch Summen von Primzahlpotenzen, II. J. Reine Angew. Math. 206 (1961), 78–112.CrossRefGoogle Scholar
14.Vaughan, R. C.. Diophantine approximation by prime numbers I. Proc. London Math. Soc.(3) 28 (1974), 373–384.CrossRefGoogle Scholar
15.Vaughan, R. C.. Diophantine approximation by prime numbers II. Proc. London Math. Soc. (3)28 (1974), 385–401.CrossRefGoogle Scholar
16.Watson, G. L.. On indefinite quadratic forms in five variables. Proc. London Math. Soc. (3) 3 (1953), 170–181.Google Scholar