Geodesic
The circle method with weights for the representation of integers by quadratic forms
Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 10, Tome 377 (2010), pp. 91-110 
N. Niedermowwe. The circle method with weights for the representation of integers by quadratic forms. Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 10, Tome 377 (2010), pp. 91-110. http://geodesic.mathdoc.fr/item/ZNSL_2010_377_a11/
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     title = {The circle method with weights for the representation of integers by quadratic forms},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2010_377_a11/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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 non-trivial. Bibl. 9 titles.

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[9] N. Niedermowwe, Zeros of Forms with $S$-unit argument, D. Phil. Thesis, University of Oxford, 2009