Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 13
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Wooley, Trevor D. 2015. Multigrade efficient congruencing and Vinogradov's mean value theorem. Proceedings of the London Mathematical Society, Vol. 111, Issue. 3, p. 519.

    Parsell, S. T. and Wooley, T. D. 2013. Exceptional Sets for Diophantine Inequalities. International Mathematics Research Notices,

    Parsell, Scott T. 2012. HUA-TYPE ITERATION FOR MULTIDIMENSIONAL WEYL SUMS. Mathematika, Vol. 58, Issue. 02, p. 209.

    Spencer, Craig V. and Wooley, Trevor D. 2012. Diophantine inequalities and quasi-algebraically closed fields. Israel Journal of Mathematics, Vol. 191, Issue. 2, p. 721.

    Wooley, Trevor 2012. Vinogradov's mean value theorem via efficient congruencing. Annals of Mathematics, Vol. 175, Issue. 3, p. 1575.

    Preobrazhenskii, S. N. 2011. New estimate in Vinogradov’s mean-value theorem. Mathematical Notes, Vol. 89, Issue. 1-2, p. 277.

    Преображенский, Сергей Николаевич and Preobrazhenskii, Sergei Nikolaevich 2011. Новая оценка в теореме о среднем И. М. Виноградова. Математические заметки, Vol. 89, Issue. 2, p. 285.

    Elliott, P. D. T. A. 2010. The value distribution of additive arithmetic functions on a line. Journal für die reine und angewandte Mathematik (Crelles Journal), Vol. 2010, Issue. 642,

    Kawada, Koichi and Wooley, Trevor D. 2010. DAVENPORT’S METHOD AND SLIM EXCEPTIONAL SETS: THE ASYMPTOTIC FORMULAE IN WARING’S PROBLEM. Mathematika, Vol. 56, Issue. 02, p. 305.

    Wooley, Trevor D. 2004. A light-weight version of Waring's problem. Journal of the Australian Mathematical Society, Vol. 76, Issue. 03, p. 303.

    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.


On Vinogradov's mean value theorem

  • Trevor D. Wooley (a1)
  • DOI:
  • Published online: 01 February 2010

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).

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *