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On Vinogradov's mean value theorem

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  26 February 2010

Trevor D. Wooley
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, yi ≥ P for 1 ≥ i ≥ s. 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

Secondary: 11P05: Waring's problem and variants
Type
Research Article
Information
Mathematika , Volume 39 , Issue 2 , December 1992 , pp. 379 - 399
Copyright
Copyright © University College London 1992

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References

1.Baker, R. C.. Diophantine Inequalities. L.M.S. Monographs, New Series (Oxford, 1986).Google Scholar
2.Hardy, G. H. and Littlewood, J. E.. Some problems of “Partitio Numerorum”: IV. Math. Zeit., 12 (1922), 161–188.CrossRefGoogle Scholar
3.Heath-Brown, D. R.. Weyl's inequality, Hua's inequality, and Waring's problem. J. hand. Math. Soc (2), 38 (1988), 216–230.Google Scholar
4.Hua, L.-K.. On Waring's problem. Quart. J. Math. Oxford, 9 (1938), 199–202.CrossRefGoogle Scholar
5.Hua, L.-K.. An improvement of Vinogradov's mean value theorem and several applications. Quart. J. Math. Oxford, 20 (1949), 48–61.CrossRefGoogle Scholar
6.Hua, L.-K.. Additive Theory of Prime Numbers (Providence, 1965).Google Scholar
7.Karatsuba, A. A.. The mean value of the modulus of a trigonometric sum. Izv. Akad. Nauk SSSR, 37 (1973), 1203–1227.Google Scholar
8.Linnik, Yu. V.. On Weyl's sums. Mat. Sbornik (Rec. Math.), 12 (1943), 23–39.Google Scholar
9.Stechkin, S. B.. On mean values of the modulus of a trigonometric sum. Trudy Mat. Inst. Steklov., 134 (1975), 283–309.Google Scholar
10.Turina, O. V.. A new estimate for a trigonometric integral of I. M. Vinogradov. Izv. Akad. Nauk SSSR, Ser. Mat., 51 (1987), No. 2. Translated in Math. USSR Izvestiya, 30 (1988), 2, 337–351.Google Scholar
11.Vaughan, R. C.. The Hardy-Littlewood Method (Cambridge University Press, 1981).Google Scholar
12.Vaughan, R. C.. On Waring's problem for cubes. J. Reine Angew. Math., 365 (1986), 122–170.Google Scholar
13.Vaughan, R. C.. On Waring's problem for smaller exponents, II. Mathematika, 33 (1986), 6–22.CrossRefGoogle Scholar
14.Vinogradov, I. M.. New estimates for Weyl sums. Dokl. Akad. Nauk SSSR, 8 (1935), 195–198.Google Scholar
15.Vinogradov, I. M.. The method of trigonometrical sums in the theory of numbers. Trav. Inst. Steklov, 23 (1947).Google Scholar
16.Walfisz, A. Z.. Weylsche Exponentialsummen in der neueren Zahlentheorie. Math. Forsch., XV (Berlin, 1963).Google Scholar
17.Wooley, T. D.. Large improvements in Waring's problem. Annals of Math., 135 (1992), 131–164.CrossRefGoogle Scholar
18.Wooley, T. D.. On Vinogradov's mean value theorem, II. Mich. Math. J. to appear.Google Scholar