# 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 $\theta<1/2$ without further hypotheses. The proof uses a recent result by Wooley on the cubic Vinogradov system.
Keywords :
Type de document :
Article dans une revue
Forum Mathematicum, De Gruyter, 2016, 28 (2), pp.403-404. <https://www.degruyter.com/view/j/form>. <10.1515/forum-2014-0216>

https://hal.archives-ouvertes.fr/hal-01464788
Contributeur : Olivier Robert <>
Soumis le : vendredi 10 février 2017 - 14:55:16
Dernière modification le : mardi 7 mars 2017 - 10:25:23

### Citation

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>

### Métriques

Consultations de la notice