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On Waring's Problem for Cubes and Smooth Weyl Sums
Author(s) -
Brüdern Jörg,
Wooley Trevor D.
Publication year - 2001
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611500012624
Subject(s) - mathematics , combinatorics , prime (order theory) , moment (physics) , order (exchange) , upper and lower bounds , pure mathematics , mathematical analysis , physics , finance , classical mechanics , economics
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 > 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.