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15 March 2013 Vinogradov’s mean value theorem via efficient congruencing, II
Trevor D. Wooley
Duke Math. J. 162(4): 673-730 (15 March 2013). DOI: 10.1215/00127094-2079905

Abstract

We apply the efficient congruencing method to estimate Vinogradov’s integral for moments of order 2s, with 1sk21. Thereby, we show that quasi-diagonal behavior holds when s=o(k2), and we obtain near-optimal estimates for 1s14k2+k and optimal estimates for sk21. In this way we come halfway to proving the main conjecture in two different directions. There are consequences for estimates of Weyl type and in several allied applications. Thus, for example, the anticipated asymptotic formula in Waring’s problem is established for sums of s kth powers of natural numbers whenever s2k22k8 (k6).

Citation

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Trevor D. Wooley. "Vinogradov’s mean value theorem via efficient congruencing, II." Duke Math. J. 162 (4) 673 - 730, 15 March 2013. https://doi.org/10.1215/00127094-2079905

Information

Published: 15 March 2013
First available in Project Euclid: 15 March 2013

zbMATH: 1312.11066
MathSciNet: MR2912712
Digital Object Identifier: 10.1215/00127094-2079905

Subjects:
Primary: 11L15
Secondary: 11L07, 11P05, 11P55

Rights: Copyright © 2013 Duke University Press

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Vol.162 • No. 4 • 15 March 2013
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