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

Abstract

Suppose $k⩾3$ 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 $r⩾2$, we have the asymptotic formula \begin{equation*} \sum_{n_1,\cdots,n_r\leqslant x}τ_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+α_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 $k⩾4$ 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.

22 pages