On the average sum of the kth divisor function over values of quadratic polynomials

Let F(x)∈Z[x1,x2,…,xn]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(x) \in \mathbb {Z}[x_1 , x_2 ,\ldots , x_n ]$$\end{document}, n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 3$$\end{document}, be an n-variable quadratic polynomial with a nonsingular quadratic part. Using the circle method we derive an asymptotic formula for the sum Σk,F(X;B)=∑x∈XB∩ZnτkF(x);\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Sigma _{k,F}(X; {{\mathcal {B}}})=\sum _{\mathbf{x}\in X{\mathcal {B}}\cap {\mathbb {Z}}^{n}}\tau _{k}\left( F(\mathbf{x})\right) ; \end{aligned}$$\end{document}for X tending to infinity, where B⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}\subset {\mathbb {R}}^n$$\end{document} is an n-dimensional box such that minx∈XBF(x)≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min \nolimits _{\mathbf{x}\in X{\mathcal {B}}}F(\mathbf{x})\ge 0$$\end{document} for all sufficiently large X, and τk(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{k}(\cdot )$$\end{document} is the kth divisor function for any integer k≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 2$$\end{document}.