On the average sum of the $k$-th divisor function over values of quadratic polynomials

Abstract

Let $F(\mathbf{x})\in \mathbb{Z}[x_1,x_2,\dots ,x_n]$ be a quadratic polynomial in $n\ge 3$ variables with a nonsingular quadratic part. Using the circle method we derive an asymptotic formula for the sum $${\Sigma }_{k,F}(X;\mathcal{B})=\sum _{\mathbf{x}\in X\mathcal{B}\cap {\mathbb{Z}}^n}{\tau }_k\left(F(\mathbf{x})\right),$$ for $X$ tending to infinity, where $\mathcal{B}\subset {\mathbb{R}}^n$ is an $n$-dimensional box such that $\underset{\mathbf{x}\in X\mathcal{B}}{\min }F(\mathbf{x})\ge 0$ for all sufficiently large $X$, and ${\tau }_k(\cdot )$ is the $k$-th divisor function for any integer $k\ge 2$.

this article supersedes arXiv:1703.10546