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

TL;DR

This paper derives an asymptotic formula for the average sum of the $k$-th divisor function over values of quadratic polynomials in multiple variables using the circle method, extending understanding of divisor sums in polynomial values.

Contribution

It introduces a novel application of the circle method to quadratic polynomials in multiple variables for divisor sum asymptotics, covering cases with non-negative polynomial values.

Findings

Asymptotic formula for divisor sums over quadratic polynomial values

Extension of circle method to multivariable quadratic forms

Results applicable for large X and non-negative polynomial values

Abstract

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