On the high-th power mean of one kind general Kloosterman sums

  • Wenpeng ZhangEmail author
  • Jiayuan Hu
Original Paper


The main purpose of this paper is using the analytic methods and a relation between the two-term cubic exponential sums and general Kloosterman sums to study the computational problem of one kind high-th power mean of general Kloosterman sums for some special non-principal character \(\chi \bmod p\), and give four exact computational formulae for them. As applications of these results, we obtained four interesting asymptotic formulae for the 6th, 8th, 10th and 12th power mean of general Kloosterman sums with a special character \(\chi \bmod p\).


Kloosterman sums High-th power mean Identity Asymptotic formula 

Mathematics Subject Classification

11L03 11L05 



The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.


  1. 1.
    Apostol, Tom M.: Introduction to Analytic Number Theory. Springer, New York (1976)zbMATHGoogle Scholar
  2. 2.
    Estermann, T.: On Kloostermann’s sums. Mathematica 8, 83–86 (1961)zbMATHGoogle Scholar
  3. 3.
    Chowla, S.: On Kloosterman’s sums. Norkse Vid. Selbsk. Fak. Frondheim 40, 70–72 (1967)zbMATHGoogle Scholar
  4. 4.
    Kloosterman, H.D.: On the representation of numbers in the form \(ax^2 +by^2+cz^2 +dt^2\). Acta Math. 49, 407–464 (1926)CrossRefGoogle Scholar
  5. 5.
    Iwaniec, H.: Topics in classical automorphic forms. Grad. Stud. Math. 17, 61–63 (1997)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Salié, H.: Uber die Kloostermanschen Summen \(S(u, v;q)\). Math. Z. 34, 91–109 (1931)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Livnė, R.: Motivic orthogonal two-dimension representations of Gal\(\left(\overline{{\mathbf{Q}}}/{\mathbf{Q}}\right)\). Israel J. Math 92, 149–156 (1995)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Perter, C., Top, J., van der Vlugt, M.: The Hasse zeta-function of a \(K3\) surface related to the number of words of weight 5 in the melas codes. J. Reine Angew. Math. 432, 151–176 (1992)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Hulek, K., Spandaw, J., van Geemen, B., van Straten, D.: The modularity of the Barth-Nieto quintic and its relatives. Adv. Geom. 1, 263–289 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Katz, Nicholas M.: Estimates for nonsingular multiplicative character sums. Int. Math. Res. Not. 7, 333–349 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Malyshev, A.V.: A generalization of Kloosterman sums and their estimates, (in Russian). Vestnik Leningrad University 15, 59–75 (1960)MathSciNetGoogle Scholar
  12. 12.
    Zhang, W.P.: The fourth and sixth power mean of the classcial Kloosterman sums. J. Num. Theory 131, 228–238 (2011)Google Scholar
  13. 13.
    Zhang, W.P.: On the general Kloosterman sums and its fourth power mean. J. Num. Theory 104, 156–161 (2004)Google Scholar
  14. 14.
    Zhang, W.P.: On the fourth power mean of the general Kloosterman sums. Indian J. Pure Appl. Math. 35, 237–242 (2004)Google Scholar
  15. 15.
    Li, J.H., Liu, Y.N.: Some new identities involving Gauss sums and general Kloosterman sums. Acta Mathematica Sinica (Chinese Series) 56, 413–416 (2013)Google Scholar
  16. 16.
    Birch, B.J.: How the number of points of an elliptic curve over a fixed prime field varies. J. Lond. Math. Soc. 43, 57–60 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Chowla, S., Cowles, J., Cowles, M.: On the number of zeros of diagonal cubic forms. J. Num. Theory 9, 502–506 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Duke, W., Iwaniec, H.: A relation between cubic exponential and Kloosterman sums. Contemp. Math. 143, 255–258 (1993)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l. 2017

Authors and Affiliations

  1. 1.School of MathematicsNorthwest UniversityXi’anPeople’s Republic of China

Personalised recommendations