On moments of the error term of the multivariable kth divisor functions

Abstract

Suppose $k⩾3$ is an integer. Let ${\tau }_k\left(n\right)$ 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 $\sum _{n_1,\cdots ,n_r⩽x}{\tau }_k(n_1\cdots n_r)=x^r\sum _{\ell =0}^{r(k-1)}{d}_{r,k,\ell }{(logx)}^{\ell }+O\left(x^{r-1+{\alpha }_k+\varepsilon }\right),$where $d_{r,k,\ell }$ and $0<{\alpha }_k<1$ are computable constants. In this paper we study the mean square of ${\Delta }_{r,k}\left(x\right)$ and give upper bounds for $k⩾4$ and an asymptotic formula for the mean square of ${\Delta }_{r,3}\left(x\right)$. We also get an upper bound for the third power moment of ${\Delta }_{r,3}\left(x\right)$. Moreover, we study the first power moment of ${\Delta }_{r,3}\left(x\right)$ and then give a result for the sign changes of it.