Bounds for the cubic Weyl sum |
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| Authors: | D R Heath-Brown |
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| Institution: | 1.Mathematical Institute,University of Oxford,Oxford,UK |
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| Abstract: | Subject to the abc-conjecture, we improve the standard Weyl estimate for cubic exponential sums in which the argument is a
quadratic irrational. Specifically. we show that
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?n \leqslant N e( an3 ) << e, aN\tfrac57 + e \sum\limits_{n \leqslant N} {e\left( {\alpha {n^3}} \right){ \ll_{\varepsilon, \alpha }}{N^{\tfrac{5}{7} + \varepsilon }}} |
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