# zbMATH — the first resource for mathematics

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
##### Keywords:
bound; exponential sum; mean value theorem; exponent pair
Full Text: