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

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.

### MSC:

 11L15 Weyl sums 11L07 Estimates on exponential sums

### Keywords:

exponential sums; Vinogradov’s method; exponent pairs
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### References:

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