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    Vaughan, R. C. 1998. Hardy's legacy to number theory. Journal of the Australian Mathematical Society, Vol. 65, Issue. 02, p. 238.


    Boklan, Kent D. 1994. The asymptotic formula in Waring's problem. Mathematika, Vol. 41, Issue. 02, p. 329.


    Wooley, Trevor D. 1993. Corrigendum: On Vinogradov's mean value theorem. Mathematika, Vol. 40, Issue. 01, p. 152.


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On Vinogradov's mean value theorem

  • Trevor D. Wooley (a1)
  • DOI: http://dx.doi.org/10.1112/S0025579300015102
  • Published online: 01 February 2010
Abstract

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).

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2.G. H. Hardy and J. E. Littlewood . Some problems of “Partitio Numerorum”: IV. Math. Zeit., 12 (1922), 161188.

17.T. D. Wooley . Large improvements in Waring's problem. Annals of Math., 135 (1992), 131164.

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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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