Higher moments of the error term in the divisor problem

It is proved that, if k ≥ 2 is a fixed integer and 1 ? H ≤ (1/2)X, then $$ \int_{X - H}^{X + H} {\Delta _k^4 \left( x \right) } dx \ll _\varepsilon X^\varepsilon \left( {HX^{{{\left( {2k - 2} \right)} \mathord{\left/ {\vphantom {{\left( {2k - 2} \right)} k}} \right. \kern-\nulldelimiterspace} k}} + H^{{{\left( {2k - 3} \right)} \mathord{\left/ {\vphantom {{\left( {2k - 3} \right)} {\left( {2k + 1} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2k + 1} \right)}}} X^{{{\left( {8k - 8} \right)} \mathord{\left/ {\vphantom {{\left( {8k - 8} \right)} {\left( {2k + 1} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2k + 1} \right)}}} } \right), $$ where Δ _k (x) is the error term in the general Dirichlet divisor problem. The proof uses a Voronoï-type formula for Δ _k (x), and the bound of Robert-Sargos for the number of integers when the difference of four kth roots is small. The size of the error term in the asymptotic formula for the mth moment of Δ₂(x) is also investigated.