On the Fourth Power Moment of the Error Term for the Divisor Problem with Congruence Conditions

Let $d(n;{\mathrm{\ell }}_1,M_1,{\mathrm{\ell }}_2,M_2)$ denote the number of factorizations $n=n_1{n}_2$, where each of the factors $n_i\in \mathbb{N}$ belongs to a prescribed congruence class ${\mathrm{\ell }}_i{M}_i(i=1,2)$. Let $\mathrm{\Delta }(x;{\mathrm{\ell }}_1,M_1,{\mathrm{\ell }}_2,M_2)$ be the error term of the asymptotic formula of $\sum _{n⩽x}d(n;{\mathrm{\ell }}_1,M_1,{\mathrm{\ell }}_2,M_2)$. In this paper, we establish an asymptotic formula of the fourth power moment of $\mathrm{\Delta }(M_1{M}_{2}x;{\mathrm{\ell }}_1,M_1,{\mathrm{\ell }}_2,M_2)$ and prove that \begin{equation*} \int_1^T\Delta^4(M_1M_2x;\ell_1,M_1,\ell_2,M_2)$\mathrm{d}$x=\frac{1}{32\pi^4}C_4\Big(\frac{\ell_1}{M_1},\frac{\ell_2}{M_2}\Big) T^2+O(T^{2-\vartheta_4+\varepsilon}), \end{equation*} with ${\vartheta }_4=1\mathrm{/}8$, which improves the previous value ${\theta }_4=3\mathrm{/}28$ of K. Liu.