Some exponent pairs by Vinogradov’s method. (Quelques paires d’exposants par la méthode de Vinogradov.) (French) Zbl 1021.11025

Robert, Olivier

J. Théor. Nombres Bordx. 14, No. 1, 271-285 (2002).

Estimates for exponential sums \(\sum_{m\sim M}e(f(m))\) depend on conditions on the \(k\)th derivative of \(f\), which should not be too large. If the relevant values of \(k\) are small, then one uses van der Corput’s method, whereas Vinogradov’s method is more powerful for large values of \(k\). The author studies the “junction” of these methods for the values \(k=9, 10, 11\). Using recent advances in both methods, he obtains the following new exponent pairs: \((1/615, 1- 9/615)\) and \((1/915, 1-10/915)\). These are applied to the estimation of Riemann’s zeta-function near the 1-line.

Reviewer: Matti Jutila (Turku)

MSC:

11L15	Weyl sums
11L07	Estimates on exponential sums

Keywords:

exponential sums; Vinogradov’s method; exponent pairs

References:

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