Journal of Mathematical Sciences

, Volume 171, Issue 6, pp 753–764 | Cite as

The circle method with weights for the representation of integers by quadratic forms

  • N. Niedermowwe

When attacking Diophantine counting problems by the circle method, the use of smoothly weighted counting functions has become commonplace to avoid technical difficulties. It can, however, be problematic to then recover corresponding results for the unweighted number of solutions.

This paper looks at quadratic forms in four or more variables representing an integer. We show how an asymptotic formula for the number of unweighted solutions in an expanding region can be obtained despite applying a weighted version of the circle method. Moreover, by carefully choosing the weight, the resulting error term is made nontrivial. Bibliography: 9 titles.


Error Term Quadratic Form Technical Difficulty Asymptotic Formula Mathematical Institute 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    T. D. Browning and R. Dietmann, “On the representation by quadratic forms,” Proc. Lond. Math. Soc., 96, No. 3, 389–416 (2008).MATHMathSciNetGoogle Scholar
  2. 2.
    G. H. Hardy and J. E. Littlewood, “Some problems of “Partitio Numerorum,” I: A new solution of Waring’s problem, in: Göttinger Nachrichten (1920), pp. 33–54.Google Scholar
  3. 3.
    D. R. Heath-Brown, “Cubic forms in ten variables,” Proc. London Math. Soc., 47, No. 3, 225–257 (1983).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    D. R. Heath-Brown, “A new form of the circle method, and its application to quadratic forms,” J. reine angew. Math., 481, 149–206 (1996).MATHMathSciNetGoogle Scholar
  5. 5.
    H. Kloosterman, “On the representation of numbers in the form ax 2 + by 2 + cz 2 + dt 2,” Acta Math., 49, 407–464 (1926).CrossRefMathSciNetGoogle Scholar
  6. 6.
    A. V. Malyshev, “On the weighted number of integer points on a quadratic,” Zap. Nauchn. Semin. LOMI, 1, 1–30 (1968).MathSciNetGoogle Scholar
  7. 7.
    B. Z. Moroz, “Distribution of integer points on multidimensional hyperboloids and cones,” Zap. Nauchn. Semin. LOMI, 1, 31–41 (1968).Google Scholar
  8. 8.
    N. Niedermowwe, ‘The density of S-integral points in projective space with respect to a quadratic,” Acta Arith., 142, 145–156 (2010).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    N. Niedermowwe, “Zeros of forms with S-unit argument,” D. Phil. Thesis, University of Oxford (2009).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Mathematical Institute, Oxford UniversityOxfordUK

Personalised recommendations