A New Minor-Arcs Estimate for Number Fields

  • Morley Davidson
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 467)

Abstract

We re-examine Körner’s number field version of a minor-arcs estimate introduced by Vinogradov into the study of Waring’s function G(k). Dependencies on both k and the degree n of the number field are reduced via improved mean value estimates for Weyl sums over ‘smooth’ algebraic integers.

Keywords

E211 Larg 

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References

  1. [B]
    Birch, B. J., Waring’s problem in algebraic number fields, Proc. Cambr. Phil. Soc. 57 (1961), 449–459.MathSciNetMATHCrossRefGoogle Scholar
  2. [D]
    Davidson, M., On Siegel’s Conjecture in Waring’s Problem,To appear.Google Scholar
  3. [E]
    Eda, Y., On Waring’s problem in an algebraic number field, Rev. Colombiana Math. 9 (1975), 29–73.MathSciNetMATHGoogle Scholar
  4. [K1]
    Körner, O., Über das Waringsche Problem in algebraischen Zahlkörpern, Math. Ann. 144 (1961), 224–238.MathSciNetMATHCrossRefGoogle Scholar
  5. [K2]
    Körner, O., Über Mittlewerte trigonometrischer Summen and ihre Anwendung in algebraischen Zahikörpern 147 (1962), 205–239.Google Scholar
  6. [M1]
    Mitsui, T., Generalized prime number theorem, Jap. Journ. Math. 26 (1956), 1–42.MathSciNetGoogle Scholar
  7. [M2]
    Mitsui, T., On the Goldbach problem in an algebraic number field I., J. Math. Soc. Japan 12 (1960), 290–324.MathSciNetCrossRefGoogle Scholar
  8. [R]
    Ramanujam, C. P., Sums of m-th powers in p-adic rings, Mathematika 10 (1963), 137–146.MathSciNetMATHCrossRefGoogle Scholar
  9. [S]
    Siegel, C. L., Generalization of Waring’s problem to algebraic number fields, American J. of Math. 66 (1944), 122–136.MATHCrossRefGoogle Scholar
  10. [T]
    Thanigasalam, K., Some new estimates for G(k) in Waring’s problem, Acta. Arith. 42 (1982), 73–78.MathSciNetMATHGoogle Scholar
  11. [V1]
    Vaughan, R. C., The Hardy-Littlewood method, Cambridge University Press, Cambridge, 1981.MATHGoogle Scholar
  12. [V2]
    Vaughan, R. C., A new iterative method in Waring’s problem, Acta Math. 162 (1989), 1–71.MathSciNetMATHCrossRefGoogle Scholar
  13. [Vi]
    Vinogradov, I. M., The method of trigonometrical sums in the theory of numbers, Interscience publishers, 1947.Google Scholar
  14. [W]
    Wang, Y., Diophantine Equations and Inequalities in Algebraic Number Fields, Springer-Verlag, Berlin Heidelberg, 1991.MATHGoogle Scholar
  15. [Wo]
    Wooley, T. D., Large improvements in Waring’s problem, Ann. Math. 135 (1992), 131–164.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Morley Davidson
    • 1
  1. 1.Dept. of Mathematics and Computer ScienceKent State UniversityKentUSA

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