On the Seventh Power Moment of $\Delta(x)$
On the Seventh Power Moment of $\Delta(x)$
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.
Comment: 18 pages
- China University of Mining and Technology China (People's Republic of)
asymptotic formula, Mathematics - Number Theory, \(\zeta (s)\) and \(L(s, \chi)\), higher-power moment, Asymptotic results on arithmetic functions, Dirichlet divisor problem
asymptotic formula, Mathematics - Number Theory, \(\zeta (s)\) and \(L(s, \chi)\), higher-power moment, Asymptotic results on arithmetic functions, Dirichlet divisor problem
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