Skip to main content
SpringerLink
Log in

Analytic Properties of Multiple Zeta-Functions in Several Variables

  • Conference paper
Number Theory

Part of the book series: Developments in Mathematics ((DEVM,volume 15))

Abstract

We report several recent results on analytic properties of multiple zeta-functions, mainly in several variables, such as the analytic continuation, the asymptotic behaviour, the location of singularities, and the recursive structure. Some results presented in this paper have never been published before.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

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

    MathSciNet  MATH  Google Scholar 

  2. S. Akiyama and H. Ishikawa, On analytic continuation of multiple L-functions and related zeta-functions, in “Analytic Number Theory”, C. Jia and K. Matsumoto (eds.), Developments in Math. Vol. 6, Kluwer, 2002, pp. 1–16.

    Google Scholar 

  3. S. Akiyama and Y. Tanigawa, Multiple zeta values at non-positive integers, Ramanujan J. 5 (2001), 327–351.

    Article  MathSciNet  MATH  Google Scholar 

  4. T. M. Apostol and T. H. Vu, Dirichlet series related to the Riemann zeta function, J. Number Theory 19 (1984), 85–102.

    Article  MathSciNet  MATH  Google Scholar 

  5. T. Arakawa and M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J. 153 (1999), 189–209.

    MathSciNet  MATH  Google Scholar 

  6. T. Arakawa and M. Kaneko, On multiple L-values, J. Math. Soc. Japan, 56 (2004), 967–991.

    MathSciNet  MATH  Google Scholar 

  7. D. Bowman and D. M. Bradley, Multiple polylogarithms: a brief survey, in “q-series with Applications to Combinatorics, Number Theory and Physics”, B. C. Berndt and K. Ono (eds.), Contem. Math. vol 291, Amer. Math. Soc, 2001, pp. 71–92.

    Google Scholar 

  8. M. de Crisenoy, Values at T-tuples of negative integers of twisted multivariable zeta series associated to polynomials of several variables, Compositio Math., to appear.

    Google Scholar 

  9. D. Essouabri, Singularités des séries de Dirichlet associées à des polynômes de plusieurs variables et applications à la théorie analytique des nombres, Thése, Univ. Henri Poincaré-Nancy I, 1995.

    Google Scholar 

  10. D. Essouabri, Singularités des séries de Dirichlet associées à des polynômes de plusieurs variables et applications en théorie analytique des nombres, Ann. Inst. Fourier 47 (1997), 429–483.

    MATH  MathSciNet  Google Scholar 

  11. H. Hida, Elementary Theory of L-functions and Eisenstein Series, London Math. Soc. Student Text 26, Cambridge Univ. Press, 1993.

    Google Scholar 

  12. J. G. Huard, K. S. Williams and N. Y. Zhang, On Tornheim’s double series, Acta Arith. 75 (1996), 105–117.

    MathSciNet  MATH  Google Scholar 

  13. H. Ishikawa, On analytic properties of a multiple L-function, in “Analytic Extension Formulas and their Applications”, S. Saitoh et al. (eds.), Soc. Anal. Appl. Comput. Vol. 9, Kluwer, 2001, pp. 105–122.

    Google Scholar 

  14. H. Ishikawa, A multiple character sum and a multiple L-function, Arch. Math. 79 (2002), 439–448.

    Article  MATH  MathSciNet  Google Scholar 

  15. H. Ishikawa and K. Matsumoto, On the estimation of the order of Euler-Zagier multiple zeta-functions, Illinois J. Math. 47 (2003), 1151–1166.

    MathSciNet  MATH  Google Scholar 

  16. K. Kamano, Multiple zeta values at non-positive integers and a generalization of Lerch’s formula, Tokyo J. Math. to appear.

    Google Scholar 

  17. M. Katsurada, An application of Mellin-Barnes’ type integrals to the mean square of Lerch zeta-functions, Collect. Math. 48 (1997), 137–153.

    MATH  MathSciNet  Google Scholar 

  18. B. Lichtin, Poles of Dirichlet series and D-modules, in “Théorie des Nombres/Number Theory”, Proc. Intern. Number Theory Conf. (Laval 1987), J.-M. De Koninck and C. Levesque (eds.), Walter de Gruyter, 1989, pp. 579–594.

    Google Scholar 

  19. B. Lichtin, The asymptotics of a lattice point problem associated to a finite number of polynomials I, Duke math. J. 63 (1991), 139–192; II, ibid. 77 (1995), 699–751.

    Article  MATH  MathSciNet  Google Scholar 

  20. B. Lichtin, Volumes and lattice points — proof of a conjecture of L. Ehrenpreis, in “Singularities, Lille 1991”, London Math. Soc. Lect. Note Vol. 201, J.-P. Brasselet (ed.), Cambridge Univ. Press, 1994, pp. 211–250.

    Google Scholar 

  21. B. Lichtin, Asymptotics determined by pairs of additive polynomials, Compos. Math. 107 (1997), 233–267.

    Article  MATH  MathSciNet  Google Scholar 

  22. K. Matsumoto, Asymptotic series for double zeta, double gamma, and Hecke L-functions, Math. Proc. Cambridge Phil. Soc. 123 (1998), 385–405; Corrigendum and addendum, ibid, 132 (2002), 377–384.

    Article  MATH  Google Scholar 

  23. K. Matsumoto, Asymptotic expansions of double gamma-functions and related remarks, in “Analytic Number Theory”, C. Jia and K. Matsumoto (eds.), Developments in Math. Vol. 6, Kluwer, 2002, pp. 243–268.

    Google Scholar 

  24. K. Matsumoto, On the analytic continuation of various multiple zeta-functions, in “Number Theory for the Millennium II, Proc. of the Millennial Conference on Number Theory” M. A. Bennett et al. (eds.), A K Peters, 2002, pp. 417–440.

    Google Scholar 

  25. K. Matsumoto, Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series, Nagoya Math. J. 172 (2003), 59–102.

    MATH  MathSciNet  Google Scholar 

  26. K. Matsumoto, The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions I, J. Number Theory 101 (2003), 223–243.

    Article  MATH  MathSciNet  Google Scholar 

  27. K. Matsumoto, The analytic continuation and the asymptotic behaviour of certain multiple zeta-functions II, in “Analytic and Probabilistic Methods in Number Theory”, Proc. 3rd Intern. Conf. in Honour of J. Kubilius, (Palanga, Lithuania, 2001), A. Dubickas et al. (eds.), TEV, Vilnius, 2002, pp. 188–194.

    Google Scholar 

  28. K. Matsumoto, On Mordell-Tornheim and other multiple zeta-functions, in “Proceedings of the Session in Analytic Number Theory and Diophantine Equations”, D. R. Heath-Brown and B. Z. Moroz (eds.), Bonner Math. Schriften Nr. 360, Univ. Bonn, 2003, n. 25, 17pp.

    Google Scholar 

  29. K. Matsumoto, Functional equations for double zeta-functions, Math. Proc. Cambridge Philos. Soc. 136 (2004), 1–7.

    Article  MATH  MathSciNet  Google Scholar 

  30. K. Matsumoto and Y. Tanigawa, The analytic continuation and the order estimate of multiple Dirichlet series, J. Théorie des Nombres de Bordeaux 15 (2003), 267–274.

    MathSciNet  MATH  Google Scholar 

  31. K. Matsumoto and H. Tsumura, Generalized multiple Dirichlet series and generalized multiple polylogarithms, preprint.

    Google Scholar 

  32. H. Mellin, Eine Formel für den Logarithmus transcendenter Funktionen von endlichem Geschlecht, Acta Soc. Sci. Fenn. 29, no. 4 (1900), 49pp.

    Google Scholar 

  33. H. Mellin, Die Dirichlet’schen Reihen, die zahlentheoretischen Funktionen und die unendlichen Produkte von endlichem Geschlecht, Acta Math. 28 (1904), 37–64.

    Article  MATH  MathSciNet  Google Scholar 

  34. L. J. Mordell, On the evaluation of some multiple series, J. London Math. Soc. 33 (1958), 368–371.

    MATH  MathSciNet  Google Scholar 

  35. T. Shintani, On evaluation of zeta functions of totally real algebraic number fields at non-negative integers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), 393–417.

    MATH  MathSciNet  Google Scholar 

  36. T. Shintani, On a Kronecker limit formula for real quadratic fields, ibid. 24 (1977), 167–199.

    MATH  MathSciNet  Google Scholar 

  37. M. V. Subbarao and R. Sitaramachandrarao, On some infinite series of L. J. Mordell and their analogues, Pacific J. Math. 119 (1985), 245–255.

    MathSciNet  MATH  Google Scholar 

  38. L. Tornheim, Harmonic double series, Amer. J. Math. 72 (1950), 303–314.

    MATH  MathSciNet  Google Scholar 

  39. H. Tsumura, On some combinatorial relations for Tornheim’s double series, Acta Arith. 105 (2002), 239–252.

    Article  MATH  MathSciNet  Google Scholar 

  40. H. Tsumura, On alternating analogues of Tornheim’s double series, Proc. Amer. Math. Soc. 131 (2003), 3633–3641.

    Article  MATH  MathSciNet  Google Scholar 

  41. H. Tsumura, Evaluation formulas for Tornheim’s type of alternative double series, Math. Comput. 73 (2003), 251–258.

    MathSciNet  Google Scholar 

  42. H. Tsumura, Multiple harmonic series related to multiple Euler numbers, J. Number Theory 106 (2004), 155–168.

    Article  MATH  MathSciNet  Google Scholar 

  43. H. Tsumura, On Mordell-Tornheim zeta values, Proc. Amer. Math. Soc. 133 (2005), 2387–2393.

    Article  MATH  MathSciNet  Google Scholar 

  44. H. Tsumura, On a class of combinatorial relations for the Mordell-Tornheim triple series, preprint.

    Google Scholar 

  45. H. Tsumura, On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function, preprint.

    Google Scholar 

  46. H. Tsumura, Certain functional relations for the double harmonic series related to the double Euler numbers, J. Austral. Math. Soc. to appear.

    Google Scholar 

  47. Maoxiang Wu, On analytic continuation of Mordell-Tornheim and Apostol-Vu L-functions (in Japanese), Master Thesis, Nagoya University, 2003.

    Google Scholar 

  48. D. Zagier, Values of zeta functions and their applications, in “First European Congress of Mathematics, Vol.II Invited Lectures (Part 2)”, A. Joseph et al. (eds.), Progress in Math. Vol. 120, Birkhäuser, 1994, pp. 497–512.

    Google Scholar 

  49. Jianqiang Zhao, Analytic continuation of multiple zeta-functions, Proc. Amer. Math. Soc. 128 (2000), 1275–1283.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan

    Kohji Matsumoto

Authors
  1. Kohji Matsumoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Northwest University, Xi’an, P.R. China

    Wenpeng Zhang

  2. Nagoya University, Nagoya, Japan

    Yoshio Tanigawa

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer Science + Business Media, Inc.

About this paper

Cite this paper

Matsumoto, K. (2006). Analytic Properties of Multiple Zeta-Functions in Several Variables. In: Zhang, W., Tanigawa, Y. (eds) Number Theory. Developments in Mathematics, vol 15. Springer, Boston, MA. https://doi.org/10.1007/0-387-30829-6_11

Download citation

Publish with us

Policies and ethics

Search

Navigation