# On the fourth derivative test for exponential sums

Abstract : We give an upper bound for the exponential sum $\sum_{m=1}^Mexp(2i\pi f(m))$ where $f$ is a real-valued function whose fourth derivative has the order of magnitude $\lambda>0$ small. Van der Corput's classical bound, in terms of $M$ and $\lambda$ only, involves the exponent $1/14$. We show how this exponent may be replaced by any $\theta<1/12$ without further hypotheses. The proof uses a recent result by Wooley on the cubic Vinogradov system.
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Journal articles
Forum Mathematicum, De Gruyter, 2016, 28 (2), pp.403-404. 〈https://www.degruyter.com/view/j/form〉. 〈10.1515/forum-2014-0216〉

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Olivier Robert. On the fourth derivative test for exponential sums. Forum Mathematicum, De Gruyter, 2016, 28 (2), pp.403-404. 〈https://www.degruyter.com/view/j/form〉. 〈10.1515/forum-2014-0216〉. 〈hal-01464788〉

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