Skip to main content
SpringerLink
Log in

Construction of Approximations to Zeta-Values

  • Conference paper
Diophantine Approximation

Part of the book series: Developments in Mathematics ((DEVM,volume 16))

  • 1392 Accesses

Abstract

Polylogarithmic functions are defined by series

$$ L_k \left( z \right) = \sum\limits_{v = 1}^\infty {\frac{{z^v }} {{v^k }}} , k \geqslant 1. $$

Due to equalities Lk;(1) = ζ(k), k ≥ 2, they play an important role in study of arithmetic properties of Riemann zeta-function ζ(s) at integer points.

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 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. Gutnik, L.A.: On the irrationality of certain quantities involving ζ(3). Usp. Mat. Nauk 34, 190 (1979) (in Russian); Acta Arith. 42, 255–264 (1983)

    MATH  MathSciNet  Google Scholar 

  2. Gutnik, L.A.: On linear independence over ℚ of dilogaritms at rational points. Usp. Mat. Nauk 37, 179–180 (1982) (in Russian); Russ. Math. Surv. 37, 176–177 (1982)

    MATH  MathSciNet  Google Scholar 

  3. Gutnik, L.A.: On the rank over ℚ of some real matrices. VINITI, 1984, No 5736-84, pp. 1–29 (in Russian)

    Google Scholar 

  4. Hessami-Pilehrood, T.G.: Lower bound of some linear form. Mat. Zametki 66, 617–623 (1999) (in Russian)

    MathSciNet  Google Scholar 

  5. Hessami Pilehrood, T.G.: Arithmetic properties of values of hypergeometric functions. Ph. D. thesis, Moscow University, Moscow, Russia (1999)

    Google Scholar 

  6. Hessami Pilehrood, T.G.: Linear independence of vectors with polylogarithmic coordinates. Vestn. Most Univ., Ser. I, 9 (6), 54–56 (1999); Mosc. Univ. Math. Bull. 54 (6), 54–56 (1999)

    MathSciNet  Google Scholar 

  7. Luke, Yu.L.: Mathematical Functions and Their Approximations. Academic Press, New York (1975)

    Google Scholar 

  8. Nesterenko, Yu.: A few remarks on ζ(3). Mat. Zametki [Math. Notes] 59, 865–880 (1996)

    MathSciNet  Google Scholar 

  9. Nesterenko, Yu.: Integral identities and constructions of approximations to zeta-values. J. Theor. Nombres Bordx. 15, 535–550 (2003)

    MATH  MathSciNet  Google Scholar 

  10. Nikishin, E.M.: On irrationality of values of functions F(x, s). Mat. Sb. 109, 410–417 (1979); Math. USSR Sb. 37, 381–388 (1980)

    MathSciNet  Google Scholar 

  11. Rhin, G., Viola, C.: On a permutation group related to ζ(2). Acta Arith. 77, 23–56 (1966)

    MathSciNet  Google Scholar 

  12. Rhin, G., Viola, C.: The group structure for ζ(3). Acta Arith. 97, 269–293 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Rivoal, T.: Propriétés diophantiennes des valeours de la fonction zêta de Riemann aux entiers impairs. Thèse de doctorat, l’Université de Caen, Caen, France (2001)

    Google Scholar 

  14. Rivoal, T.: La fonction Zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs. C. R. Acad. Sci. Paris, Sér. I Math. 331, 267–270 (2000)

    MATH  MathSciNet  Google Scholar 

  15. Slater, L.J.: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966)

    Google Scholar 

  16. Whittaker, E.F., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1927)

    Google Scholar 

  17. Zudilin, V.V.: One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational. Usp. Mat. Nauk 56, 149–150 (2001); Russ. Math. Surv. 56, 774–776 (2001)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow Lomonosov State University, Vorobiovy Gory, GSP-2, 119992, Moscow, Russia

    Yuri V. Nesterenko

Authors
  1. Yuri V. Nesterenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Philipps-Universität Marburg, Marburg, Germany

    Hans Peter Schlickewei

  2. Universität Wien, Vienna, Austria

    Klaus Schmidt

  3. Technische Universität Graz, Graz, Austria

    Robert F. Tichy

Additional information

To Professor Wolfgang M. Schmidt on the occasion of his 70th birthday

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag

About this paper

Cite this paper

Nesterenko, Y.V. (2008). Construction of Approximations to Zeta-Values. In: Schlickewei, H.P., Schmidt, K., Tichy, R.F. (eds) Diophantine Approximation. Developments in Mathematics, vol 16. Springer, Vienna. https://doi.org/10.1007/978-3-211-74280-8_16

Download citation

Publish with us

Policies and ethics

Search

Navigation