This paper develops a new form of the circle method, in which the ‘Kloosterman refinement’ is built in, with the special feature that there are no exponential sums involving \(a^{-1} \pmod q\) which one would previously have expected. The number of solutions of the equation under investigation is expressed, in closed form, as \[ \sum_{c \in \mathbb{Z}^n} \sum_{q = 1}^\infty q^{-n} S_q({\mathbf c}) I_q({\mathbf c}), \] where \(S_q({\mathbf c})\) is a complete exponential sum to modulus \(q\), and \(I_q({\mathbf c})\) is an integral. The contribution from \({\mathbf c} = \mathbf{0}\) is essentially the classical main term, while other \({\mathbf c}\) produce remainder terms. The starting point for the analysis is an identity of W. Duke, J. Friedlander and H. Iwaniec [Invent. Math. 112, 1-8 (1993; Zbl 0765.11038)].
The new method is applied to the problem of representing a large integer by a quadratic form, reproducing well-known results for forms of rank 4 or more. Moreover the problem of representations of zero is also investigated. Here the Hasse principle is established, in quantitative form, for ranks 3 or more. Previously this had been possible by the circle method only for forms of rank 5 or more. In the case of rank 3, the main term in the asymptotic formula for the number of solutions in a sphere of radius \(P\) is of order \(P\log P\), and it is of particular interest to observe how such a leading term arises.