On the higher moments of the error term in the divisor problem

Abstract

Let $\Delta (x)$ denote the error term in the Dirichlet divisor problem. Our main results are the asymptotic formulas $${\int }_1^X{\Delta }^3(x)\mathrm{d}x=B{X}^{7/4}+O_{\epsilon }(X^{\beta +\epsilon })(B>0)$$ and $${\int }_1^X{\Delta }^4(x)\mathrm{d}x=C{X}^2+O_{\epsilon }(X^{\gamma +\epsilon })(C>0)$$ with $\beta =7/5,\gamma =23/12$. This improves on the values $\beta =47/28,\gamma =45/23$, due to K.-M. Tsang. A result on the integrals of ${\Delta }^3(x)$ and ${\Delta }^4(x)$ in short intervals is also proved.

27 pages