# The sum of multidimensional divisor function over values of quadratic polynomial

## TL;DR

This paper derives explicit asymptotic formulas for the sum of the multidimensional divisor function over values of a positive quadratic polynomial in three or more variables, using the circle method.

## Contribution

It provides the first explicit asymptotic formulas for sums involving the multidimensional divisor function over quadratic polynomial values in multiple variables.

## Key findings

- Asymptotic formulas are obtained for the sum T_{k,F}(X).
- Results apply to quadratic polynomials with non-zero discriminant.
- The circle method is effectively used to analyze these sums.

## Abstract

Let $F({\bf x})={\bf x}^tQ_m{\bf x}+\mathbf{b}^t{\bf x}+c\in\mathbb{Z}[{\bf x}]$ be a quadratic polynomial in $\ell (\ge 3 )$ variables ${\bf x} =(x_{1},...,x_{\ell})$, where $F({\bf x})$ is positive when ${\bf x}\in\mathbb{R}_{\ge 1}^{\ell}$, $Q_m\in {\rm M}_{\ell}(\mathbb{Z})$ is an $\ell\times\ell$ matrix and its discriminant $\det\left(Q_m^t+Q_m\right)\neq 0$. It gives explicit asymptotic formulas for the following sum \[ T_{k,F}(X)=\sum_{{\bf x}\in [1,X]^{\ell}\cap\mathbb{Z}^{\ell}}\tau_{k}\left(F({\bf x})\right) \] with the help of the circle method. Here $\tau_{k}(n)=\#\{(x_1,x_2,...,x_{k})\in\mathbb{N}^{k}: n=x_1x_2...x_{k}\}$ with $k\in\mathbb{Z}_{\ge 2}$ is the multidimensional divisor function.