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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 \({\Bbb Q}(\sqrt { - 3} )\) 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 X5/6, 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].

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Correspondence to D. R. Heath-Brown.

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Heath-Brown, D.R. Kummer’s conjecture for cubic Gauss sums. Isr. J. Math. 120, 97–124 (2000).

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