On the Seventh Power Moment of $\Delta(x)$

Abstract

Let $\Delta(x)$ be the error term of the Dirichlet divisor problem. In this paper, we establish an asymptotic formula of the seventh-power moment of $\Delta(x)$ and prove that \begin{equation*} \int_2^T \Delta^7(x)\mathrm{d}x= \frac{7(5s_{7;3}(d)-3s_{7;2}(d)-s_{7;1}(d))}{2816\pi^7}T^{11/4}+O(T^{11/4-\delta_7+\varepsilon}) \end{equation*} with $\delta_7=1/336,$ which improves the previous result.