  Let $F({\bf x})={\bf x}^tQ_m{\bf x}+\mathbf{b}^t{\bf x}+c\in\mathbb{Z}[{\bf
x}]$ be a quadratic polynomial in $\ell (\ge 3 )$ variables ${\bf x}
=(x_{1},...,x_{\ell})$, where $F({\bf x})$ is positive when ${\bf
x}\in\mathbb{R}_{\ge 1}^{\ell}$, $Q_m\in {\rm M}_{\ell}(\mathbb{Z})$ is an
$\ell\times\ell$ matrix and its discriminant $\det\left(Q_m^t+Q_m\right)\neq
0$. It gives explicit asymptotic formulas for the following sum \[
T_{k,F}(X)=\sum_{{\bf x}\in
[1,X]^{\ell}\cap\mathbb{Z}^{\ell}}\tau_{k}\left(F({\bf x})\right) \] with the
help of the circle method. Here
$\tau_{k}(n)=\#\{(x_1,x_2,...,x_{k})\in\mathbb{N}^{k}: n=x_1x_2...x_{k}\}$ with
$k\in\mathbb{Z}_{\ge 2}$ is the multidimensional divisor function.
