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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
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 $\varepsilon >0$.
Received: January 18, 2016
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
Linking options:
https://www.mathnet.ru/eng/tm3777https://doi.org/10.1134/S0371968517010071 https://www.mathnet.ru/eng/tm/v296/p95
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