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A new kth derivative estimate for exponential sums via Vinogradov’s mean value

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Abstract

We give a slight refinement to the process by which estimates for exponential sums are extracted from bounds for Vinogradov’s mean value. Coupling this with the recent works of Wooley, and of Bourgain, Demeter and Guth, providing optimal bounds for the Vinogradov mean value, we produce a powerful new kth derivative estimate. Roughly speaking, this improves the van der Corput estimate for k ≥ 4. Various corollaries are given, showing for example that \(\zeta \left( {\sigma + it} \right){ \ll _\varepsilon }{t^{{{\left( {1 - \sigma } \right)}^{3/2}}/2 + \varepsilon }}\) for t ≥ 2 and 0 ≤ σ ≤ 1, for any fixed ε > 0.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    D. R. Heath-Brown

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  1. D. R. Heath-Brown
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Correspondence to D. R. Heath-Brown.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Vol. 296, pp. 95–110.

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Heath-Brown, D.R. A new kth derivative estimate for exponential sums via Vinogradov’s mean value. Proc. Steklov Inst. Math. 296, 88–103 (2017). https://doi.org/10.1134/S0081543817010072

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