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

TL;DR

This paper derives an explicit asymptotic formula for the average of the divisor function over the values of a positive quadratic polynomial in three or more variables, providing new insights into divisor sums in polynomial sequences.

Contribution

It introduces a novel explicit asymptotic formula for the divisor function sum over quadratic polynomial values in multiple variables, extending previous results to higher dimensions.

Findings

Explicit asymptotic formula derived for divisor sums

Applicable to quadratic polynomials with non-zero discriminant

Enhances understanding of divisor distribution in polynomial values

Abstract

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