On moments of the error term of the multivariable k-th divisor functions

Abstract

Suppose $k\geqslant3$ is an integer. Let $\tau_k(n)$ be the number of ways $n$ can be written as a product of $k$ fixed factors. For any fixed integer $r\geqslant2$, we have the asymptotic formula \begin{equation*} \sum_{n_1,\cdots,n_r\leqslant x}\tau_k(n_1 \cdots n_r)=x^r\sum_{\ell=0}^{r(k-1)}d_{r,k,\ell}(\log x)^{\ell}+O(x^{r-1+\alpha_k+\varepsilon}), \end{equation*} where $d_{r,k,\ell}$ and $0<\alpha_k<1$ are computable constants. In this paper we study the mean square of $\Delta_{r,k}(x)$ and give upper bounds for $k\geqslant4$ and an asymptotic formula for the mean square of $\Delta_{r,3}(x)$. We also get an upper bound for the third power moment of $\Delta_{r,3}(x)$. Moreover, we study the first power moment of $\Delta_{r,3}(x)$ and then give a result for the sign changes of it.