Skip to main content

On Analytic Properties of a Multiple L-Function

  • Chapter
Analytic Extension Formulas and their Applications

Part of the book series: International Society for Analysis, Applications and Computation ((ISAA,volume 9))

Abstract

A multiple L-function is introduced and its analytic continuation is established by a simple application of the Euler-Maclaurin summation formula. Moreover the location of poles is investigated and the absolute value of the function with respect to the imaginary part of the variable is estimated.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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, An analytic continuation of multiple zeta functions and their values at non-positive integers, to appear in Acta Arith.

    Google Scholar 

  2. S. Akiyama and H. Ishikawa, On analytic continuation of multiple L-functions and related zeta-functions, to appear in Analytic Number Theory.,C.Jia and K.Matsumoto (eds.), Kluwer Academic Publishers.

    Google Scholar 

  3. 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 

  4. F.V. Atkinson, The mean value of the Riemann zeta-function, Acta Math., 81 (1949), 353376.

    Google Scholar 

  5. J.M. Borwein, D.M. Bradley, D.J. Broadhurst and P. Lisonék, Combinatrial aspects of multiple zeta values, Electron.J.Combin.,5(1998),No.1, Research Paper38, 12 pp.

    Google Scholar 

  6. D.J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett., B 393 (1997), No 3–4, 403–412.

    Google Scholar 

  7. S. Egami, Introduction to multiple zeta function,Lecture Note at Niigata University (in Japanese).

    Google Scholar 

  8. A.B.Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett., 5 (1998), no. 4, 497–516.

    MathSciNet  Google Scholar 

  9. H. Ishikawa, A Multiple character sum and a multiple L-function, preprint.

    Google Scholar 

  10. A. Ivié, The Riemann zeta-Function, Wiley-Interscience Publication, New York, 1985.

    Google Scholar 

  11. M. Katsurada, An application of Mellin-Barnes type integrals to the mean square of L-functions, Riet. Matem. Rink., 38 (1998) 98–112.

    MathSciNet  Google Scholar 

  12. M. Katsurada and K. Matsumoto, Asymptotic expansions of the mean values of Dirichlet L-functions. Math. Z., 208 (1991), 23–39.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Katsurada and K. Matsumoto, Explicite formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions I, Math. Scand., 78 (1996), 161–177.

    MathSciNet  MATH  Google Scholar 

  14. K. Matsumoto, Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series, preprint.

    Google Scholar 

  15. Y. Motohashi, A note on the mean value of the zeta and L-functions. I, Proc. Japan Acad., Ser. A Math. Sci., 61 (1985), 222–224.

    MathSciNet  MATH  Google Scholar 

  16. Y. Ohno, A generalization of the duality and sum formulas on the multiple zeta values J.of Number Th. 74 (1999) 39–43.

    Google Scholar 

  17. D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II, Sirkhäuser, (1994), 210–220

    Google Scholar 

  18. D. Zagier, Periods of modular forms, traces of Hecke operators, and multiple zeta values, Research into automorphic forms and L functions (in Japanese) RIMS Kokyuroku, 843 (1993), 162–170.

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Ishikawa, H. (2001). On Analytic Properties of a Multiple L-Function. In: Saitoh, S., Hayashi, N., Yamamoto, M. (eds) Analytic Extension Formulas and their Applications. International Society for Analysis, Applications and Computation, vol 9. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3298-6_8

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-3298-6_8

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-4854-0

  • Online ISBN: 978-1-4757-3298-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics