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.