The sum of multidimensional divisor function over values of quadratic polynomial

Abstract

Let $F(\mathbf{x})={\mathbf{x}}^t{Q}_m\mathbf{x}+{\mathbf{b}}^t\mathbf{x}+c\in \mathbb{Z}[\mathbf{x}]$ be a quadratic polynomial in $\ell (\ge 3)$ variables $\mathbf{x}=(x_1,...,x_{\ell })$, where $F(\mathbf{x})$ is positive when $\mathbf{x}\in {\mathbb{R}}_{\ge 1}^{\ell }$, $Q_m\in {\mathrm{M}}_{\ell }(\mathbb{Z})$ is an $\ell \times \ell$ matrix and its discriminant $\det \left(Q_m^t+Q_m\right)\ne 0$. It gives explicit asymptotic formulas for the following sum \[ T_{k,F}(X)=\sum_{{\bf x}\in [1,X]^{\ell}\cap\mathbb{Z}^{\ell}}τ_{k}\left(F({\bf x})\right) \] with the help of the circle method. Here ${\tau }_k(n)=\#\{(x_1,x_2,...,x_k)\in {\mathbb{N}}^k:n=x_1{x}_2...x_k\}$ with $k\in {\mathbb{Z}}_{\ge 2}$ is the multidimensional divisor function.

arXiv comment16 pages

References in corpus (1)

• Le problème des diviseurs pour des formes binaires de degré 4