On quadratic forms in four variables

Question

Let $F$ be a non-singular integral quadratic form in four variables. Then a result of Heath-Brown from the 90's states for $m \to \infty$, $$|\{ x \in \mathbb{Z}^4 \,:\, F(x) = m \}| = C_F\sigma(F,m)m + O_{F,\varepsilon}(m^{\frac{3}{4} + \varepsilon}),$$ where $C_F$ is a positive constant depending only on $F$ and $\sigma(F,m)$ is the singular series.

I am interested to know, whether up till today there are improvements on the error rate of this result. More precisely, is there any hope to improve the error rate to $O_{F,\varepsilon}(m^{\frac{1}{2} + \varepsilon})$. If this is unknown, what is the conjectured error term?

For the application I have in mind, it would be already interesting to understand the case $$F(x) = x_1^2 + x_2^2 + x_3^2 + x_4^2.$$ Are there any results for this case or any other specific quadratic form?

Answer 1

It follows from Deligne's bound for the Hecke eigenvalues of weight $2$ holomorphic cusp forms (which is really Eichler's theorem in this special case) that the error term is $O_{F,\varepsilon}(m^{\frac{1}{2} + \varepsilon})$. Indeed, as proved by Siegel, the main term is the Eisenstein contribution of the underlying theta series (which is a holomorphic modular form of weight $2$), so the error term is the cuspidal contribution.

In particular, for $F(x) = x_1^2 + x_2^2 + x_3^2 + x_4^2$ the error term is zero, since the main term in this case reproduces Jacobi's well-known formula $8\sum_{\substack{d\mid m\\4\nmid d}}d$ for the number of representations.