Higher moments of the error term in the divisor problem

Abstract

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 Δk(x) is the error term in the general Dirichlet divisor problem. The proof uses a Voronoi-type formula for Δk(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.