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{...
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| Format: | Journal article |
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2000
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| _version_ | 1797055456917585920 |
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| author | Heath-Brown, D |
| author_facet | Heath-Brown, D |
| author_sort | Heath-Brown, D |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-06T19:11:00Z |
| format | Journal article |
| id | oxford-uuid:16c3966e-61cb-499d-839c-559b3794fcff |
| institution | University of Oxford |
| last_indexed | 2024-03-06T19:11:00Z |
| publishDate | 2000 |
| record_format | dspace |
| spelling | oxford-uuid:16c3966e-61cb-499d-839c-559b3794fcff2022-03-26T10:33:13ZKummer's conjecture for cubic Gauss sumsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:16c3966e-61cb-499d-839c-559b3794fcffMathematical Institute - ePrints2000Heath-Brown, DLet $S(X,l)=\sum_{N(c)\leq X}\tilde{g}(c)\Lambda(c)(\frac{c}{|c|})^l$ 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. |
| spellingShingle | Heath-Brown, D Kummer's conjecture for cubic Gauss sums |
| title | Kummer's conjecture for cubic Gauss sums |
| title_full | Kummer's conjecture for cubic Gauss sums |
| title_fullStr | Kummer's conjecture for cubic Gauss sums |
| title_full_unstemmed | Kummer's conjecture for cubic Gauss sums |
| title_short | Kummer's conjecture for cubic Gauss sums |
| title_sort | kummer s conjecture for cubic gauss sums |
| work_keys_str_mv | AT heathbrownd kummersconjectureforcubicgausssums |