Arithmetic functions on average over values of quadratic polynomial

Nianhong Zhou

Published 2016-11-30Version 1

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 $\det\left(Q_m^t+Q_m\right)\neq 0$. This paper uses the circle method to study the following sum \[ T(F,g;X)=\sum_{{\bf x}\in [1,X]^{\ell}\cap\mathbb{Z}^{\ell}}g\left(F({\bf x})\right), \] where $g$ is an arithmetic function which satisfies proper conditions. The present paper takes the divisor function $\tau$ and the Mangoldt function $\Lambda$ as examples. It gives explicit asymptotic formulas for $T(F,\tau;X)$ and $T(F,\Lambda;X)$ with error terms. The paper finally gives some applications for the results.