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

Abstract

Let $d(n;{\ell }_1,M_1,{\ell }_2,M_2)$ 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 ${\ell }_i\mathrm{mod}{M}_i(i=1,2)$. Let $\Delta (x;{\ell }_1,M_1,{\ell }_2,M_2)$ be the error term of the asymptotic formula of $\sum _{n⩽x}d(n;{\ell }_1,M_1,{\ell }_2,M_2)$. In this paper, we establish an asymptotic formula of the fourth power moment of $\Delta (M_1{M}_{2}x;{\ell }_1,M_1,{\ell }_2,M_2)$ and prove that \begin{equation*} \int_1^TΔ^4(M_1M_2x;\ell_1,M_1,\ell_2,M_2)\mathrm{d}x=\frac{1}{32π^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 ${\vartheta }_4=1/8$, which improves the previous value ${\theta }_4=3/28$ of K. Liu.

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