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?

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  • $\begingroup$ Here is the original paper. core.ac.uk/display/77442361 Jayce Getz refined the error: doi.org/10.1112/jlms.12130 $\endgroup$ – MyNinthAccount Jan 30 at 11:15
  • $\begingroup$ @MyNinthAccount: As far as I understand, Getz is only proving results for the case where the sum is over F(x) = 0. $\endgroup$ – Constantin K Jan 30 at 11:35
  • $\begingroup$ You should ask separate questions as separate questions, don't make your question a moving target. If the original question has been answered to your satisfaction, then accept the answer; and if you want to ask a different question, do that separately. $\endgroup$ – Alex B. Feb 4 at 10:24
  • $\begingroup$ Ok – here is the new question: mathoverflow.net/questions/351915/… $\endgroup$ – Constantin K Feb 4 at 13:45

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.

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