Higher moments of the error term in the divisor problem

It is proved that, if $k\ge2$ is a fixed integer and $1 \ll H \le X/2$, then $$ \int_{ X-H}^{ X+H}\Delta^4_k(x)\d x \ll_\epsilon X^\epsilon\Bigl(HX^{ (2k-2)/k} + H^{ (2k-3)/(2k+1)}X^{ (8k-8)/(2k+1)}\Bigr), $$ where $\Delta_k(x)$ is the error term in the general Dirichlet divisor problem. The proof uses the Vorono{ \"\i}--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. We also investigate the size of the error term in the asymptotic formula for the $m$-th moment of $\Delta_2(x)$.

Categories: math.NT