Skip to main content
SpringerLink
Log in

Renormalization of multiple q-zeta values

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we shall define the renormalization of the multiple q-zeta values (MqZV) which are special values of multiple q-zeta functions ζ q (s 1, ..., s d ) when the arguments are all positive integers or all non-positive integers. This generalizes the work of Guo and Zhang (Renormalization of Multiple Zeta Values, arxiv: math/0606076v3). We show that our renormalization process produces the same values if the MqZVs are well-defined originally and that these renormalizations of MqZV satisfy the q-stuffle relations if we use shifted-renormalizations for all divergent ζ q (s 1, ..., s d ) (i.e., s 1 ≤ 1). Moreover, when q ↑ 1 our renormalizations agree with those of Guo and Zhang.

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. Euler, L.: Meditationes circa singulare serierum genus. Novi. Comm. Acad. Sci. Petropolitanae, 20, 140–186 (1775)

    Google Scholar 

  2. Broadhurst, D. J., Kreimer, D.: Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops. Phys. Lett. B, 393, 403–412 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Goncharov, A. B., Manin, Yu. I.: Multiple zeta-motives and moduli spaces M 0,n . Compositio Math., 140, 1–4 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Hoffman, M. E.: Multiple harmonic series. Pacific J. Math., 152(2), 275–290 (1992)

    MATH  MathSciNet  Google Scholar 

  5. Le, T. Q. T., Murakami, J.: Kontsevich’s integral for the Homfly polynomial and relations between values of the multiple zeta functions. Topology Appl., 62, 193–206 (1995)

    Article  MathSciNet  Google Scholar 

  6. Zagier, D.: Values of zeta function and their applications. Proceedings of the First European Congress of Mathematics, 2, 497–512 (1994)

    MathSciNet  Google Scholar 

  7. Zhao, J.: Analytic continuation of multiple zeta functions. Proc. of Amer. Math. Soc., 128, 275–1283 (1999)

    Google Scholar 

  8. Akiyama, S., Egami, S., Tanigawa, Y.: Analytic continuation of multiple zeta-functions and their values at non-positive integers. Acta Arith., 98, 107–116 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ebrahimi-Fard, K., Guo, L.: Rota-Baxter algebras and multiple zeta values. To appear in Integers, Preprint at arxiv: math.NT/0601558

  10. Guo, L., Zhang, B.: Renormalization of multiple zeta values. J. Algebra, 319(9), 3770–3809 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Guo, L., Zhang, B.: Differential Birkhoff decomposition and renormalization of multiple zeta values. J. Number Theory, 128(8), 2318–2339 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ihara, K., Kaneko, M., Zagier, D.: Derivation and double shuffle relations for multiple zeta values. Compositio Math., 142, 307–338 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Zhao, J.: Multiple-q zeta functions and multiple-q polylogarithms. Ramanujan J., 14(2), 189–221 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kaneko, M., Kurokawa, N., Wakayama, M.: A variation of Euler’s approach to values of the Riemann zeta function. Kyushu J. Math., 57, 175–192 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hoffman, M. E.: Quasi-shuffle products. J. Algebraic Combin, 11, 49–68 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Zudlin, W.: Algebraic relations for multiple zeta values, (Russian). Uspekhi Mat. Nauk., 58(1), 3–32 (2003), translation in Russian Math. Survey, 58(1), 1–29 (2003)

    MathSciNet  Google Scholar 

  17. Whittaker, E. T., Watson, G. N.: A Course of Modern Analysis, 4th ed., Cambridge University Press, 1996

  18. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions, Dover, New York, 1964

    MATH  Google Scholar 

  19. Artin, E.: The Gamma Function, New York, Holt, Rinehart and Winston, 1964. Reprinted in Exposition by Emil Artin: A Selection, ed. by Michael Rosen. History of Mathematics 30, Sources Subseries. Amer. Math. Soc./London Math. Soc., 2007

  20. Knopp, K.: Theory and Application of Infinite Series, 2nd ed., Blackie and Son Limited, London and Glasgow, 1951

    MATH  Google Scholar 

  21. Manchon, D.: Hopf algebras, from basics to applications to renormalization. Comptes-rendus des Rencontres mathmématiques de Glanon 2001 arxiv: math.QA/0408405

  22. Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem. Comm. Math. Phys., 210(1), 249–273 (2000)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Eckerd College, St. Petersburg, FL, 33711, USA

    Jianqiang Zhao

Authors
  1. Jianqiang Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianqiang Zhao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhao, J. Renormalization of multiple q-zeta values. Acta. Math. Sin.-English Ser. 24, 1593–1616 (2008). https://doi.org/10.1007/s10114-008-6646-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-008-6646-x

Keywords

MR(2000) Subject Classification

Search

Navigation