Kummer's conjecture for cubic Gauss sums

It is shown that the normalized cubic Gauss sums for integers c = 1 ((mod 3)) of the field Q root -3 satisfy [GRAPHICS] for every I E Z and any E > 0. This improves on the estimate established by Heath-Brown and Patterson [4] in demonstrating the uniform distribution of the cubic Gauss sums around the unit circle. When l = 0 it is conjectured that the above sum is asymptotically of order X-5/6, so that the upper bound is essentially best possible. The proof uses a cubic analogue of the author's mean value estimate for quadratic character sums [3].