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.
Document type :
Journal articles
Forum Mathematicum, De Gruyter, 2016, 28 (2), pp.403-404. 〈https://www.degruyter.com/view/j/form〉. 〈10.1515/forum-2014-0216〉
Liste complète des métadonnées

Cited literature [5 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-01464788
Contributor : Olivier Robert <>
Submitted on : Tuesday, June 12, 2018 - 2:40:10 PM
Last modification on : Friday, October 26, 2018 - 10:47:28 AM
Document(s) archivé(s) le : Thursday, September 13, 2018 - 10:39:32 PM

File

fourth-test-Forum.pdf
Files produced by the author(s)

Identifiers

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〉

Share

Metrics

Record views

182

Files downloads

24