Skip to main content
SpringerLink
Log in

Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values of the q-logarithm function, with natural choice of~q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’s series for γ, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler’s constant. The main tools are Euler-type integrals and hypergeometric series.

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.

Similar content being viewed by others

References

  1. Bailey, W.N.: Generalized Hypergeometric Series, Cambridge Math. Tracts Vol. 32 Cambridge Univ. Press Cambridge (1935) 2nd reprinted edition New York–London Stechert-Hafner (1964)

  2. Berndt, B.C., Bowman, D.C.: Ramanujan’s short unpublished manuscript on integrals and series related to Euler’s constant. Constructive, Experimental, and Nonlinear Analysis (Limoges, 1999) CMS Conf. Proc. M. Thera Amer. Math. Soc. Providence, RI, 27, 19–27 (2000)

  3. Borwein, P.: On the irrationality of \(\sum\frac1q^n+r\). J. Number Theory. 37, 253–259 (1991)

    Google Scholar 

  4. Catalan, E.: Sur la constante d’Euler et la fonction de Binet. Liouville J. I(3), 209–240 (1875)

    MATH  Google Scholar 

  5. Gosper, Jr. R.W.: Personal communication (7 May 2002); Item 120: Accelerating series. HAKMEM Artificial Intelligence Memo No. 239 Massachusetts Institute of Technology, A. I. Laboratory (1972); http://www.inwap.com/pdp10/hbaker/hakmem/hakmem.html

  6. Rosser, J.B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Illinois J. Math. 6, 64–94 (1962)

    MATH  MathSciNet  Google Scholar 

  7. Sebah, P.: Personal communication (17 December 2002)

  8. Sondow, J. Criteria for irrationality of Euler’s constant. Proc. Amer. Math. Soc. 131, 3335–3344 (2003)

    Google Scholar 

  9. Sondow, J.: A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler’s constant; E-print math.NT/0211075 (November 2002)

  10. Vacca, G.: A new series for the Eulerian constant γ = .577 γ. Quart. J. Pure Appl. Math. 41, 363–364 (1910)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. 209 West 97th Street, New York, NY, 10025, United States

    Jonathan Sondow

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

    Wadim Zudilin

Authors
  1. Jonathan Sondow
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wadim Zudilin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonathan Sondow.

Additional information

Dedication: To Leonhard Euler on his 300th birthday.

2000 Mathematics Subject Classification Primary—11Y60; Secondary—11J72, 33C20, 33D15

The work of the second author is supported by an Alexander von Humboldt research fellowship

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sondow, J., Zudilin, W. Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper. Ramanujan J 12, 225–244 (2006). https://doi.org/10.1007/s11139-006-0075-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-006-0075-1

Keywords

Search

Navigation