Skip to main content
SpringerLink
Log in

On the 2k-th power mean of \( \frac{{L'}} {L}(1,\chi ) \) with the weight of Gauss sums

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

The main purpose of this paper is to study the hybrid mean value of \( \frac{{L'}} {L}(1,\chi ) \) and Gauss sums by using the estimates for trigonometric sums as well as the analytic method. An asymptotic formula for the hybrid mean value \( \sum\limits_{\chi \ne \chi _0 } {|\tau (\chi )||\frac{{L'}} {L}(1,\chi )|^{2k} } \) of \( \frac{{L'}} {L} \) and Gauss sums will be proved using analytic methods and estimates for trigonometric sums.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Tom M. Apostol: Introduction to Analytic Number Theory. Springer-Verlag, New York, 1976, pp. 160–162.

    MATH  Google Scholar 

  2. Pan Chengdong and Pan Chengbiao: Element of the Analytic Number Theory. Science Press, Beijing, 1991, pp. 243–248.

    Google Scholar 

  3. K. Ireland and M. Rosen: A Classical Introduction to Modern Number Theory. Springer-Verlag, New York, 1982, pp. 88–91.

    MATH  Google Scholar 

  4. Yi Yuan and Zhang Wenpeng: On the first power mean of Dirichlet L-functions with the weight of Gauss sums. Journal of Systems Science and Mathematical Sciences 20 (2000), 346–351.

    MATH  Google Scholar 

  5. Yi Yuan and Zhang Wenpeng: On the 2k-th Power mean of Dirichlet L-function with the weight of Gauss sums. Advances in Mathematics 31 (2002), 517–526.

    Google Scholar 

  6. H. Davenport: Multiplicative Number Theory. Springer-Verlag, New York, 1980.

    MATH  Google Scholar 

  7. Zhang Wenpeng: A new mean value formula of Dirichlet’s L-function. Science in China (Series A) 35 (1992), 1173–1179.

    MATH  Google Scholar 

  8. Liu Huaning and Zhang Xiaobeng: On the mean value of \( \frac{{L'}} {L}(1,\chi ) \). Journal of Mathematical Analysis and Applications 320 (2006), 562–577.

    Article  MATH  MathSciNet  Google Scholar 

  9. C. L. Siegel: Über die Klassenzahl quadratischer Zahlkörper. Acta. Arith. 1 (1935), 83–86.

    MATH  Google Scholar 

  10. Pan Chengdong and Pan Chengbiao: The Elementary Number Theory. Peking University Press, Beijing, 2003.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Center for Basic Science, Xi’an Jiaotong University, Xi’an, Shaanxi, P.R. China, 710049

    Dongmei Ren & Yuan Yi

Authors
  1. Dongmei Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuan Yi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dongmei Ren.

Additional information

Cordially dedicated to editor V.Dlab

This work is supported by N.S.F. (10601039) of P.R. China.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ren, D., Yi, Y. On the 2k-th power mean of \( \frac{{L'}} {L}(1,\chi ) \) with the weight of Gauss sums. Czech Math J 59, 781–789 (2009). https://doi.org/10.1007/s10587-009-0047-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10587-009-0047-x

Keywords

MSC 2000

Search

Navigation