Mean square of the error term in the asymmetric multidimensional divisor problem

Abstract

Let $\mathbf{a}=(a_1,\cdots,a_k)$ denote a $k$-tuple of positive integers such that $a_1 \leq a_2 \leq\break \cdots \leq a_k$. We put $d(\mathbf{a};n)=\sum_{n_1^{a_1}\cdots n_k^{a_k}=n}1$ and let $\Delta(\mathbf{a};x)$ be the error term of the corresponding asymptotic formula for the summatory function of $d(\mathbf{a};n)$. In this paper we show an asymptotic formula of the mean square of $\Delta(\mathbf{a};x)$ under a certain condition. Moreover, when $k$ equals $2$ or $3$, we give unconditional asymptotic formulas for these mean squares.