Kummer’s conjecture for cubic Gauss sums

Abstract

It is shown that the normalized cubic Gauss sums for integers c ≡ 1 ((mod 3)) of the field ${\Bbb Q}(\sqrt { - 3} )$ satisfy

$${\sum\limits_{N(c) \leqslant X} {\tilde g(c)\Lambda (c)\left( {\frac{c}{{\left| c \right|}}} \right)} ^l} \ll {}_\varepsilon {X^{5/6 + \varepsilon }} + \left| l \right|{X^{3/4 + \varepsilon }},$$

for every l ∈ ℤ and any ε > 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].