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


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 
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Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Mathematical Institute, Oxford UniversityOxfordUK

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