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ON WARING'S PROBLEM FOR CUBES AND SMOOTH WEYL SUMS

Published online by Cambridge University Press:  12 January 2001

JÖRG BRÜDERN  and
TREVOR D. WOOLEY
JÖRG BRÜDERN
Affiliation:
Mathematisches Institut A, Universität Stuttgart, Postfach 80 11 40, D-70511 Stuttgart, Germany, bruedern@mathematik.uni-stuttgart.de
TREVOR D. WOOLEY
Affiliation:
Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, MI 48109-1109, USA, wooley@math.lsa.umich.edu
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Abstract

Non-trivial estimates for fractional moments of smooth cubic Weyl sums are developed. Complemented by bounds for such sums of use on both the major and minor arcs in a Hardy--Littlewood dissection, these estimates are applied to derive an upper bound for the $s$th moment of the smooth cubic Weyl sum of the expected order of magnitude as soon as $s\ge 7.691$. Related arguments demonstrate that all large integers $n$ are represented as the sum of eight cubes of natural numbers, all of whose prime divisors are at most $\exp (c(\log n\log \log n)^{1/2})$, for a suitable positive number $c$. This conclusion improves a previous result of G. Harcos in which nine cubes are required. 1991 Mathematics Subject Classification: 11P05, 11L15, 11P55.

Keywords

Waring's problemsmooth numbersexponential sums
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 82 , Issue 1 , January 2001 , pp. 89 - 109
Copyright
2001 London Mathematical Society

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