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On Vinogradov's mean value theorem

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  26 February 2010

Trevor D. Wooley
Trevor D. Wooley
Affiliation:
Department of Mathematics, University of MichiganAnn Arbor, MI 48109-1003U.S.A..
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The object of this paper is to obtain improvements in Vinogradov's mean value theorem widely applicable in additive number theory. Let Js,k(P) denote the number of solutions of the simultaneous diophantine equations

with 1 ≥ xi, yi ≥ P for 1 ≥ i ≥ s. In the mid-thirties Vinogradov developed a new method (now known as Vinogradov's mean value theorem) which enabled him to obtain fairly strong bounds for Js,k(P). On writing

in which e(α) denotes e2πiα, we observe that

where Tk denotes the k-dimensional unit cube, and α = (α1,…,αk).

MSC classification

Secondary: 11P05: Waring's problem and variants
Type
Research Article
Information
Mathematika , Volume 39 , Issue 2 , December 1992 , pp. 379 - 399
Copyright
Copyright © University College London 1992

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