# On quadratic forms in four variables

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?

• Here is the original paper. core.ac.uk/display/77442361 Jayce Getz refined the error: doi.org/10.1112/jlms.12130 – MyNinthAccount Jan 30 at 11:15
• @MyNinthAccount: As far as I understand, Getz is only proving results for the case where the sum is over F(x) = 0. – Constantin K Jan 30 at 11:35
• 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. – Alex B. Feb 4 at 10:24
• Ok – here is the new question: mathoverflow.net/questions/351915/… – 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.