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

Abstract

Suppose that $a$ and $b$ are positive integers subject to $(a,b)=1$. For $n\in\mathbb{Z}^+$, denote by $\tau_{a,b}(n;\ell_1,M_1,l_2,M_2)$ the asymmetric two--dimensional divisor function with congruence conditions, i.e., \begin{equation*} \tau_{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 $\sum_{n\leqslant M_1^aM_2^bx}\tau_{a,b}(n;\ell_1,M_1,l_2,M_2)$. This result constitutes an enhancement upon the previous result of Zhai and Cao [16].