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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Volume 296, Pages 95–110
DOI: https://doi.org/10.1134/S0371968517010071
(Mi tm3777)
 

This article is cited in 25 scientific papers (total in 25 papers)

A new $k$th derivative estimate for exponential sums via Vinogradov's mean value

D. R. Heath-Brown

Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Oxford, UK
References:
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 $k$th derivative estimate. Roughly speaking, this improves the van der Corput estimate for $k\ge 4$. Various corollaries are given, showing for example that $\zeta (\sigma +it)\ll _{\varepsilon }t^{(1-\sigma )^{3/2}/2+\varepsilon }$ for $t\ge 2$ and $0\le \sigma \le 1$, for any fixed $\varepsilon >0$.
Funding agency Grant number
Engineering and Physical Sciences Research Council EP/K021132X/1
This work was supported by the EPSRC grant no. EP/K021132X/1.
Received: January 18, 2016
English version:
Proceedings of the Steklov Institute of Mathematics, 2017, Volume 296, Pages 88–103
DOI: https://doi.org/10.1134/S0081543817010072
Bibliographic databases:
Document Type: Article
UDC: 511.323
Language: Russian
Citation: D. R. Heath-Brown, “A new $k$th derivative estimate for exponential sums via Vinogradov's mean value”, Analytic and combinatorial number theory, Collected papers. On the occasion of the 125th anniversary of the birth of Academician Ivan Matveevich Vinogradov, Trudy Mat. Inst. Steklova, 296, MAIK Nauka/Interperiodica, Moscow, 2017, 95–110; Proc. Steklov Inst. Math., 296 (2017), 88–103
Citation in format AMSBIB
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\paper A new $k$th derivative estimate for exponential sums via Vinogradov's mean value
\inbook Analytic and combinatorial number theory
\bookinfo Collected papers. On the occasion of the 125th anniversary of the birth of Academician Ivan Matveevich Vinogradov
\serial Trudy Mat. Inst. Steklova
\yr 2017
\vol 296
\pages 95--110
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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  • This publication is cited in the following 25 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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