On the mean square of the error term for the asymmetric two-dimensional divisor problem with congruence conditions

Suppose that $a$ and $b$ are positive integers subject to $(a,b)=1$. For $n\in {\mathbb{Z}}^{+}$, denote by ${\tau }_{a,b}(n;{\mathrm{\ell }}_1,M_1,l_2,M_2)$ the asymmetric two–dimensional divisor function with congruence conditions, i.e., \begin{equation*} τ_{a,b}(n;\ell_1,M_1,l_2,M_2)=\sum_{\substack{n=n_1^an_2^b\ n_1\equiv\ell_1!!!!!\pmod{M_1}\ n_2\equiv\ell_2!!!!!\pmod{M_2}}}1. \end{equation*} In this paper, we shall establish an asymptotic formula of the mean square of the error term of the sum ${\sum }_{n⩽M_1^a{M}_2^{b}x}{\tau }_{a,b}(n;{\mathrm{\ell }}_1,M_1,l_2,M_2)$. This result constitutes an enhancement upon the previous result of Zhai and Cao [16].