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

Abstract

Let $F(\mathbf{x})={\mathbf{x}}^t{Q}_{\mathbf{x}}+{\mathbf{b}}^t\mathbf{x}+c\in \mathbb{Z}[\mathbf{x}]$ be a quadratic polynomial in $\ell (\ge 3)$ variables $\mathbf{x}=(x_1,...,x_{\ell })$, where $F(\mathbf{x})$ is positive when $\mathbf{x}\in {\mathbb{R}}_{\ge 1}^{\ell }$, $Q\in {\mathrm{M}}_{\ell }(\mathbb{Z})$ is an $\ell \times \ell$ matrix and its discriminant $\det \left(Q^t+Q\right)\ne 0$. It gives an explicit asymptotic formula for the following sum \[\sum_{{\bf x}\in [1,X]^{\ell}\cap\mathbb{Z}^{\ell}}τ\left(F({\bf x})\right), \] where $\tau$ is the divisor function.