In short, the question is for any references describing how to use the Hardy-Littlewood circle method to find an asymptotic for the number of solutions to $F(x\_1, ..., x\_s) = k$ for $(x\_1, ..., x\_s) \in \mathbb{Z}^s$, where $F$ is some indefinite integral quadratic form, and $k\neq 0$ is a fixed integer.


An old paper of Sarnak et. al. (A Proof of Siegel's Weight Formula, see [here](https://collaborate.princeton.edu/en/publications/a-proof-of-siegels-weight-formula) or [here, for a PDF](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.543.8732&rep=rep1&type=pdf)) mentions that solving such a problem can be done with the Hardy-Littlewood circle method if $s\geq 5$ (the paper mentions this in section 2, just before formula 2.2). They mention the $s=4$ case is harder, and provide a reference to a paper; I tracked down the paper they refer to, although again it seems to only cover the diagonal case. I am not very intereseted in this case; I just want to see how to solve it for $s\geq 5$ for non-diagonal forms. I found some [lecture notes](https://web.math.princeton.edu/%7Egyujino/Sarnak%202020.pdf) of a course given by Sarnack covering the case of diagonal forms for $s\geq 5,$ but I do not see how to generalize. Is there some easy way to go from knowing this result for diagonal forms to non-diagonal ones, or some other source which covers solutions to indefinite forms? The proof given by Sarnack in those notes seems to use in a fundamental way the fact that the form is diagonal, so that a certain factorization can work.



Hardy-Littlewood circle method for non-diagonal quadratic forms
The "best" way to deal with quadratic forms using the circle method is via Heath-Brown's delta symbol method.


You can read about this in detail in the paper:
Heath-Brown - A New Form of the Circle Method, and its application to Quadratic Forms


Theorem 4 in particular gives an asymptotic formula for the problem you mention. Note that Heath-Brown is also able to obtain results for 4 variables.


