Kummer's conjecture for cubic Gauss sums - Heath-Brown, D (2000)
Let $S(X,l)=\sum_{N(c)≤ X}\tilde{g}(c)Λ(c)(\frac{c}{|c|})^l$ where $\tilde{g}(c)$ is the normalized cubic Gauss sum for an integer $c\equiv 1\pmod{3}$ of the field $\mathbb{Q}(\sqrt{-3})$. It is shown that $S(X,l)\ll_{ε} X^{5/6+\ep}+|l|X^{3/4+ε}$, for every $l\in\mathbb{Z}$ and any $ε>0$. This improves on the estimate established by Heath-Brown and Patterson 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.