On the lattice point problem for ellipsoids

For k-dimensional ellipsoids E with k 9 we show that the number of lattice points in rE is approximated by the volume of rE up to an error of order O(r k\Γ2 ), as r tends to infinity. The estimate is best possible and uniform in the class of ellipsoids with bounded ratios of lengths of axes. It improves the classical estimate O(r k\Γ2k=(k+1) ) due to Landau (1915).