Kummer's conjecture for cubic Gauss sums

Heath-Brown, D. R. (2000) Kummer's conjecture for cubic Gauss sums. Israel Journal of Mathematics, 120 . pp. 97-124.

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Abstract

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 $l\in\mathbb{Z}$ and any $\varepsilon>0$ . 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.

Item Type:	Article
Subjects:	H - N > Number theory
Research Groups:	Number Theory Group
ID Code:	158
Deposited By:	Roger Heath-Brown
Deposited On:	14 Jan 2005
Last Modified:	20 Jul 2009 14:19

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