Higher Moments of the Error Term in the Divisor Problem

It is proved that, if $k\ge 2$ is a fixed integer and $1\ll H\le(1/2)X$, then
$$ \int_{X-H}^{X+H}\Delta^4_k(x)\,dx \ll_\varepsilon X^\varepsilon (HX^{(2k-2)/k}+H^{(2k-3)/(2k+1)}X^{(8k-8)/(2k+1)}), $$
where $\Delta_k(x)$ is the error term in the general Dirichlet divisor problem. The proof uses a Voronoï–type formula for $\Delta_k(x)$, and the bound of Robert–Sargos for the number of integers when the difference of four $k$th roots is small. The size of the error term in the asymptotic formula for the $m$th moment of $\Delta_2(x)$ is also investigated.