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A criterion for the fifth derivative for exponential sums. (Un critère de la dérivée cinquième pour les sommes d’exponentielles.) (French) Zbl 1027.11058
From the text: The author gives a bound \(S_M\ll_\varepsilon M^{1+\varepsilon} \lambda^{7/192}\) for \(M\gg \lambda^{-7/16}\) (and also \(S_M\ll_\varepsilon M^{1+\varepsilon} \lambda^{17/456}\) for \(M\gg \lambda^{11/19}\)) for the exponential sum \(S_M= \sum_{m=1}^M e(f(m))\) where \(f\) is a real-valued function whose fifth derivative is of a constant small size, say \(\lambda\), by means of \(M\) and \(\lambda\), improving an old result of Van der Corput \((S_M\ll M\lambda^{1/30}\) for \(M\gg \lambda^{-8/15})\). The proof relies on a mean value theorem for sixth powers of exponential sums which is treated independently in O. Robert and P. Sargos [Publ. Inst. Math., Nouv. Sér. 67(81), 14-30 (2000; Zbl 1006.11046)].
He also obtains the following result: Theorem. For all \(\varepsilon>0\), \((17/456+ \varepsilon\), \(388/456+ \varepsilon)\) is an exponent pair.

MSC:
11L07 Estimates on exponential sums
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