The explicit asymptotic formula of divisor function on average over values of quadratic polynomial

Abstract: 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.