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Monatshefte für Mathematik

, Volume 122, Issue 3, pp 265–273 | Cite as

Some remarks on Vinogradov's mean value theorem and Tarry's problem

  • Trevor D. Wooley
Article

Abstract

LetW(k, 2) denote the, least numbers for which the system of equations\(\sum\nolimits_{i = 1}^s {x_i^j = } \sum\nolimits_{i = 1}^s {y_i^j (1 \leqslant j \leqslant k)} \) has a solution with\(\sum\nolimits_{i = 1}^s {x_i^{k + 1} \ne } \sum\nolimits_{i = 1}^s {y_i^{k + 1} } \). We show that for largek one hasW(k, 2)≦1/2k2(logk+loglogk+O(1)), and moreover that whenK is large, one hasW(k, 2)≦1/2k(k+1)+1 for at least one valuek in the interval [K, K3/4+ε]. We show also that the leasts for which the expected asymptotic formula holds for the number of solutions of the above system of equations, inside a box, satisfiessk2(logk+O(loglogk).

1991 Mathematics Subject Classification

11D72 11P55 11L07 

Key words

Vinogradov's mean value theorem Tarry's problem exponential sums Hardy-Littlewood method 

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References

  1. [1]
    Baker, R. C.: Diophantine Inequalities. L.M.S. Monographs, New Series, Oxford, 1986.Google Scholar
  2. [2]
    Hardy, G. H., Wright, E. M.: An Introduction to the Theory of Number. 5th ed. Oxford: Clarendon Press. 1979.Google Scholar
  3. [3]
    Hua, L.-K.: On Tarry's problem. Quart. J. Math. (Oxford)9, 315–320 (1938).Google Scholar
  4. [4]
    Hua, L.-K.: Improvement of a result of Wright. J. London Math. Soc.24, 157–159 (1949).Google Scholar
  5. [5]
    Hua, L.-K.: Additive Theory of Prime Numbers, Providence, R.I.: Amer. Math. Soc. 1965.Google Scholar
  6. [6]
    Vaughan, R. C.: The Hardy-Littlewood Method. Cambridge: Univ. Press. 1981.Google Scholar
  7. [7]
    Vaughan, R. C.: A new iterative method in Waring's problem. Acta Math.162, 1–71 (1989).Google Scholar
  8. [8]
    Wooley, T. D.: Large improvements in Waring's problem. Ann. of Math.135, 131–164. (1992).Google Scholar
  9. [9]
    Wooley, T. D.: On Vinogradov's mean value theorem. Mathematika39, 379–399 (1992).Google Scholar
  10. [10]
    Wooley, T. D.: A note on symmetric diagonal equations. In: Pollington, A. D., Moran, W. (eds), Number Theory with an Emphasis on the Markoff Spectrum, pp. 317–321. New York: Marcel Dekker. 1993.Google Scholar
  11. [11]
    Wooley, T. D.: Quasi-diagonal behaviour in certain mean value theorems of additive number theory, J. Amer. Math. Soc.7, 221–245 (1994).Google Scholar
  12. [12]
    Wright, E. M.: On Tarry's problem. I. Quart. J. Math.. Oxford6, 261–267 (1935).Google Scholar
  13. [13]
    Wright, E. M.: Equal sums of like powers. Bull. Amer. Math. Soc.54, 755–757 (1948).Google Scholar
  14. [14]
    Wright, E. M.: The Prouhet-Lehmer problem. J. London Math. Soc.23, 279–285 (1948).Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Trevor D. Wooley
    • 1
  1. 1.Mathematics DepartmentUniversity of MichiganMichiganAnn ArborUSA

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