Journal of Mathematical Sciences

, Volume 171, Issue 6, pp 753–764

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

Article

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.

## Keywords

Error Term Quadratic Form Technical Difficulty Asymptotic Formula Mathematical Institute
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## References

1. 1.
T. D. Browning and R. Dietmann, “On the representation by quadratic forms,” Proc. Lond. Math. Soc., 96, No. 3, 389–416 (2008).
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).
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).
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).
6. 6.
A. V. Malyshev, “On the weighted number of integer points on a quadratic,” Zap. Nauchn. Semin. LOMI, 1, 1–30 (1968).
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).
9. 9.
N. Niedermowwe, “Zeros of forms with S-unit argument,” D. Phil. Thesis, University of Oxford (2009).Google Scholar