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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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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DOI: https://doi.org/10.1007/0-387-30829-6_4
Publisher Name: Springer, Boston, MA
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