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On Vinogradov's mean value theorem

Published online by Cambridge University Press:  26 February 2010

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, yiP for 1 ≥ is. 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

Type
Research Article
Copyright
Copyright © University College London 1992

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