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 Tr. Mat. Inst. Steklova, 2017, Volume 296, Pages 95–110 (Mi tm3777)

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

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.

DOI: https://doi.org/10.1134/S0371968517010071

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English version:
Proceedings of the Steklov Institute of Mathematics, 2017, 296, 88–103

Bibliographic databases:

UDC: 511.323

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, Tr. 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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\by D.~R.~Heath-Brown
\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 Tr. Mat. Inst. Steklova
\yr 2017
\vol 296
\pages 95--110
\publ MAIK Nauka/Interperiodica
\mathnet{http://mi.mathnet.ru/tm3777}
\crossref{https://doi.org/10.1134/S0371968517010071}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3640775}
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\transl
\jour Proc. Steklov Inst. Math.
\yr 2017
\vol 296
\pages 88--103
\crossref{https://doi.org/10.1134/S0081543817010072}
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• http://mi.mathnet.ru/eng/tm3777
• https://doi.org/10.1134/S0371968517010071
• http://mi.mathnet.ru/eng/tm/v296/p95

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This publication is cited in the following articles:
1. Blomer V., Bourgain J., Radziwill M., Rudnick Z., “Small Gaps in the Spectrum of the Rectangular Billiard”, Ann. Sci. Ec. Norm. Super., 50:5 (2017), 1283–1300
2. Y. Akbal, A. M. Guloglu, “Waring-Goldbach problem with Piatetski-Shapiro primes”, J. Theor. Nr. Bordx., 30:2 (2018), 449–467
3. Erdogan M.B., Shakan G., “Fractal Solutions of Dispersive Partial Differential Equations on the Torus”, Sel. Math.-New Ser., 25:1 (2019), UNSP 11
4. Pierce L.B., “The Vinogradov Mean Value Theorem [After Wooley, and Bourgain, Demeter and Guth]”, Asterisque, 2019, no. 407, 479+
5. Kumchev A., Petrov Zh., “A Hybrid of Two Theorems of Piatetski-Shapiro”, Mon.heft. Math., 189:2 (2019), 355–376
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