On the Waring–Goldbach problem for eighth and higher powers

Angel V. Kumchev; Trevor D Wooley

doi:10.1112/jlms/jdw013

On the Waring–Goldbach problem for eighth and higher powers

Angel V. Kumchev, Trevor D Wooley

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Abstract

Recent progress on Vinogradov's mean value theorem has resulted in improved estimates for exponential sums of Weyl type. We apply these new estimates to obtain sharper bounds for the hunck H(k) in the Waring-Goldbach problem. We obtain new results for all exponents k ≥ 8, and in particular establish that H(k) ≤ (4k - 2) log k + k - 7 when k is large, giving the first improvement on the classical result of Hua from the 1940s.

Original language	English
Pages (from-to)	811-824
Number of pages	14
Journal	Journal of the London Mathematical Society
Volume	93
Issue number	3
Early online date	4 May 2016
DOIs	https://doi.org/10.1112/jlms/jdw013
Publication status	Published - Jun 2016

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This is the author accepted manuscript (AAM). The final published version (version of record) is available online via Oxford University Press at http://jlms.oxfordjournals.org/content/93/3/811.abstract. Please refer to any applicable terms of use of the publisher.
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title = "On the Waring–Goldbach problem for eighth and higher powers",

abstract = "Recent progress on Vinogradov's mean value theorem has resulted in improved estimates for exponential sums of Weyl type. We apply these new estimates to obtain sharper bounds for the hunck H(k) in the Waring-Goldbach problem. We obtain new results for all exponents k ≥ 8, and in particular establish that H(k) ≤ (4k - 2) log k + k - 7 when k is large, giving the first improvement on the classical result of Hua from the 1940s.",

author = "Kumchev, {Angel V.} and Wooley, {Trevor D}",

year = "2016",

month = jun,

doi = "10.1112/jlms/jdw013",

language = "English",

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T1 - On the Waring–Goldbach problem for eighth and higher powers

AU - Kumchev, Angel V.

AU - Wooley, Trevor D

PY - 2016/6

Y1 - 2016/6

N2 - Recent progress on Vinogradov's mean value theorem has resulted in improved estimates for exponential sums of Weyl type. We apply these new estimates to obtain sharper bounds for the hunck H(k) in the Waring-Goldbach problem. We obtain new results for all exponents k ≥ 8, and in particular establish that H(k) ≤ (4k - 2) log k + k - 7 when k is large, giving the first improvement on the classical result of Hua from the 1940s.

AB - Recent progress on Vinogradov's mean value theorem has resulted in improved estimates for exponential sums of Weyl type. We apply these new estimates to obtain sharper bounds for the hunck H(k) in the Waring-Goldbach problem. We obtain new results for all exponents k ≥ 8, and in particular establish that H(k) ≤ (4k - 2) log k + k - 7 when k is large, giving the first improvement on the classical result of Hua from the 1940s.

U2 - 10.1112/jlms/jdw013

DO - 10.1112/jlms/jdw013

M3 - Article (Academic Journal)

SN - 0024-6107

VL - 93

SP - 811

EP - 824

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 3

ER -

On the Waring–Goldbach problem for eighth and higher powers

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