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A Fourth Derivative Test for Exponential Sums

Published online by Cambridge University Press:  04 December 2007

O. Robert  and
P. Sargos
O. Robert
Affiliation:
Institut Elie Cartan, Université Henri Poincaré – Nancy I, BP 239, 54 506 Vandoeuvre-lès-Nancy Cedex, France. E-mail: robert@iecn.u-nancy.fr
P. Sargos
Affiliation:
Institut Elie Cartan, Université Henri Poincaré – Nancy I, BP 239, 54 506 Vandoeuvre-lès-Nancy Cedex, France. E-mail: sargos@iecn.u-nancy.fr
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Abstract

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We give an upper bound for the exponential sum [sum ]Mm=1 e2iπf(m) in terms of M and λ, where λ is a small positive number which denotes the size of the fourth derivative of the real valued function f. The classical van der Corput's exponent 1/14 is improved into 1/13 by reducing the problem to a mean square value theorem for triple exponential sums.

Keywords

exponential sumsdiophantine systems
Type
Research Article
Information
Compositio Mathematica , Volume 130 , Issue 3 , February 2002 , pp. 275 - 292
Copyright
© 2002 Kluwer Academic Publishers