  Let $Q$ be a nondegenerate quadratic form on a vector space $V$ of even
dimension $n$ over a number field $F$. Via the circle method or automorphic
methods one can give good estimates for smoothed sums over the number of zeros
of the quadratic form whose coordinates are of size at most $X$ (properly
interpreted). For example, when $F=\mathbb{Q}$ and $\dim V>4$ Heath-Brown has
given an asymptotic of the form \begin{align} \label{HB:esti}
c_1X^{n-2}+O_{Q,\varepsilon,f}(X^{n/2+\varepsilon}) \end{align} for any
$\varepsilon>0$. Here $c_1 \in \mathbb{C}$ and $f \in
\mathcal{S}(V(\mathbb{R}))$ is a smoothing function. We refine Heath-Brown's
work to give an asymptotic of the form $$
c_1X^{n-2}+c_2X^{n/2}+O_{Q,\varepsilon,f}(X^{n/2+\varepsilon-1}) $$ over any
number field. Here $c_2 \in \mathbb{C}$. Interestingly the secondary term $c_2$
is the sum of a rapidly decreasing function on $V(\mathbb{R})$ over the zeros
of $Q^{\vee}$, the form whose matrix is inverse to the matrix of $Q$. We also
prove analogous results in the boundary case $n=4$, generalizing and refining
Heath-Brown's work in the case $F=\mathbb{Q}$.
