Secondary Terms in Asymptotics for the Number of Zeros of Quadratic Forms

Let $F$ be a non-degenerate quadratic form on an $n$-dimensional vector space $V$ over the rational numbers, and let $J$ be the symmetric matrix associated to $F$. One is interested in counting the number of zeros of the quadratic form whose coordinates are restricted in a smoothed box of size $B$. Heath-Brown gave an asymptotic for this question of the form: $c_1 B^{n-2} + O_{J,\epsilon}(B^{(n-1+\delta)/2+\epsilon})$, for any $\epsilon > 0$ and dim$V \geq 5$, where $c_1 \in \mathbb{C}$ and $\delta=0$ or $1$, according as $n$ is odd or even. For dim$V = 3$ and dim$V = 4$, Heath-Brown also gave similar asymptotics. More recently, Getz gave an asymptotic of the form: $c_1 B^{n-2} + c_2 B^{n/2} + O_{J,\epsilon}(B^{n/2+\epsilon-1})$ when $n$ is even, in which $c_2 \in \mathbb{C}$ has a pleasant geometric interpretation. We consider the case where $n$ is odd with diagonal unimodular $J$ and give an analogous asymptotic of the form: $c_1 B^{n-2} + c_2 B^{(n-1)/2} + O_{J,\epsilon}(B^{n/2+\epsilon-1})$. We use the circle method and work classically to exploit Gauss sums and find Dirichlet characters that fit into the odd degree case. We also provide an explicit description of the Dirichlet series arisen during the investigation, which is useful in applications. It turns out that the geometric interpretation of the constant $c_2$ of the asymptotic in the odd degree and even degree cases is strikingly different.