  Let $F({\bf x})={\bf x}^tQ_{\bf x}+\mathbf{b}^t{\bf x}+c\in\mathbb{Z}[{\bf
x}]$ be a quadratic polynomial in $\ell (\ge 3 )$ variables ${\bf x}
=(x_{1},...,x_{\ell})$, where $F({\bf x})$ is positive when ${\bf
x}\in\mathbb{R}_{\ge 1}^{\ell}$, $Q\in {\rm M}_{\ell}(\mathbb{Z})$ is an
$\ell\times\ell$ matrix and its discriminant $\det\left(Q^t+Q\right)\neq 0$. It
gives an explicit asymptotic formula for the following sum \[\sum_{{\bf x}\in
[1,X]^{\ell}\cap\mathbb{Z}^{\ell}}\tau\left(F({\bf x})\right), \] where $\tau$
is the divisor function.
