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 for the integral of the cube and the fourth power of $\Delta(x)$. The exponents that we obtain in the error terms, namely $\beta = {\sfrac{7}{5}}$ and $\gamma = {\sfrac{23}{12}}$, respectively, are new. They improve on the values $\beta = {\sfrac{47}{28}}, \gamma = {\sfrac{45}{23}}$, due to K.-M. Tsang. A result on integrals of $\Delta^3(x)$ and $\Delta^4(x)$ in short intervals is also proved.