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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

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Source Normalized Impact per Paper (SNIP) 2018: 0.964

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Volume 28, Issue 2


On the fourth derivative test for exponential sums

Olivier Robert
  • Université de Lyon and Université de Saint-Etienne, Institut Camille Jordan CNRS UMR 5208, 23, rue du Dr P. Michelon, F-42000, Saint-Etienne, France
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Published Online: 2015-05-21 | DOI: https://doi.org/10.1515/forum-2014-0216


We give an upper bound for the exponential sum ∑m=1,...,M exp(2iπf(m)) where f is a real-valued function whose fourth derivative has the order of magnitude λ > 0 small. Van der Corput's classical bound, in terms of M and λ only, involves the exponent 1/14. We show how this exponent may be replaced by any θ < 1/12 without further hypotheses. The proof uses a recent result by Wooley on the cubic Vinogradov system.

Keywords: Exponential sums; Vinogradov system

MSC: 11L07

About the article

Received: 2014-12-12

Published Online: 2015-05-21

Published in Print: 2016-03-01

Citation Information: Forum Mathematicum, Volume 28, Issue 2, Pages 403–404, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2014-0216.

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