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On the Hilbert-Kamke and the Vinogradov Problems in Additive Number Theory

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Number Theory

Part of the book series: Developments in Mathematics ((DEVM,volume 15))

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Abstract

This paper is survey on recent results on the system of equations x s1 + ... + x s k = βs, 1 ≤ s ≤ n, in natural number unknowns — the Hilbert-Kamke system, and one on prime unknowns — the Vinogradov system of equations. The main problem is to determine or estimate the Hardy-Littlewood function G(m) through (mean values of) trigonometrical sums and find the exponent of convergence of the associated singular integrals. We shall also state the corresponding results on the multivariate version of these problems.

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Author information

Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, M. V. Lomonosov Moscow State University, Vorobjovy Gory, 119992, Moscow, Russia

    Vladimir N. Chubarikov

Authors
  1. Vladimir N. Chubarikov
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Editor information

Editors and Affiliations

  1. Northwest University, Xi’an, P.R. China

    Wenpeng Zhang

  2. Nagoya University, Nagoya, Japan

    Yoshio Tanigawa

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Chubarikov, V.N. (2006). On the Hilbert-Kamke and the Vinogradov Problems in Additive Number Theory. In: Zhang, W., Tanigawa, Y. (eds) Number Theory. Developments in Mathematics, vol 15. Springer, Boston, MA. https://doi.org/10.1007/0-387-30829-6_4

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