Kummer's conjecture for cubic Gauss sums

Let $S(X,l)=\sum_{N(c)\leq X}\tilde{g}(c)\Lambda(c)(\frac{c}{|c|})^l$ 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 $S(X,l)\ll_{\varepsilon} X^{5/6+\ep}+|l|X^{3/4+\varepsilon}$, for every $l\in\mathbb{...