Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-11T02:09:13.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Sums of three cubes

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  26 February 2010

Trevor D. Wooley
Trevor 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

The set of integers represented as the sum of three cubes of natural numbers is widely expected to have positive density (see Hooley [7] for a discussion of this topic). Over the past six decades or so, the pursuit of an acceptable approximation to the latter statement has spawned much of the progress achieved in the theory of the Hardy-Littlewood method, so far as its application to Waring's problem for smaller exponents is concerned. Write R(N) for the number of positive integers not exceeding N which are the sum of three cubes of natural numbers.

MSC classification

Secondary: 11P05: Waring's problem and variants
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 53 - 61
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.)

References

1.Brüdern, J.. A problem in additive number theory. Math. Proc. Camb. Phil. Soc., 103 (1988), 27–33.CrossRefGoogle Scholar
2.Brüdern, J.. On Waring's problem for cubes. Math. Proc. Camb. Phil. Soc., 109 (1991), 229–256.CrossRefGoogle Scholar
3.Brüdern, J. and Wooley, T. D.. On Waring's problem for cubes and smooth Weyl sums. Proc. London Math. Soc. (3), 82 (2001), 89–109.CrossRefGoogle Scholar
4.Davenport, H.. On Waring's problem for cubes. Acta Math., 71 (1939), 123–143.CrossRefGoogle Scholar
5.Davenport, H.. Sums of three positive cubes. J. London Math. Soc., 25 (1950), 339–343.CrossRefGoogle Scholar
6.Heath-Brown, D. R.. The circle method and diagonal cubic forms. Phil. Trans. Roy. Soc. London Ser. A, 356 (1998), 673–699.CrossRefGoogle Scholar
7.Hooley, C.. On some topics connected with Waring's problem. J. reine angew. Math., 369 (1986), 110–153.Google Scholar
8.Hooley, C.. On Waring's problem. Acta Math., 157 (1986), 49–97.CrossRefGoogle Scholar
9.Hooley, C.. On Hypothesis K * in Waring's problem. Sieve methods, exponential sums and their applications in number theory (Cardiff, 1995), London Math. Soc. Lecture Note Ser., 237 (Cambridge University Press, Cambridge 1997), pp. 175–185.Google Scholar
10.Ringrose, C. J.. Sums of three cubes. J. London Math. Soc. (2), 33 (1986), 407–413.CrossRefGoogle Scholar
11.Vaughan, R. C.. Sums of three cubes. Bull. London Math. Soc., 17 (1985), 17–20.CrossRefGoogle Scholar
12.Vaughan, R. C.. On Waring's problem for cubes. J. reine angew. Math., 365 (1986), 122–170.Google Scholar
13.Vaughan, R. C.. A new iterative method in Waring's problem. Acta Math., 162 (1989), 1–71.CrossRefGoogle Scholar
14.Vaughan, R. C.. The Hardy-Littlewood Method, 2nd edition (Cambridge University Press, Cambridge, 1997).CrossRefGoogle Scholar
15.Wooley, T. D.. Breaking classical convexity in Waring's problem: Sums of cubes and quasidiagonal behaviour. Inventiones Math., 122 (1995), 421–451.CrossRefGoogle Scholar