Higher moments of the error term in the divisor problem

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It is proved that, if k≥2 is a fixed integer and 1 ≪ H ≤ X/2, then ∫ₓ₋ₕˣ⁺ʰΔ⁴ₖ(x) x ≪εXε(HX⁽²ᵏ⁻²⁾/ᵏ + H⁽²ᵏ⁻³⁾/⁽²ᵏ⁺¹⁾X⁽⁸ᵏ⁻⁸⁾/⁽²ᵏ⁺¹⁾), where Δₖ(x) is the error term in the general Dirichlet divisor problem. The proof uses the Voronoı--type formula for Δₖ(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 Δ₂(x).