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New Vacca-Type Rational Series for Euler’s Constant γ and Its “Alternating” Analog \(\ln \frac{4}{\pi }\)

  • Jonathan SondowEmail author
Chapter

Summary

We recall a pair of logarithmic series that reveals ln(4 ∕ π) to be an “alternating” analog of Euler’s constant γ. Using the binary expansion of an integer, we derive linear, quadratic, and cubic analogs for ln(4 ∕ π) of Vacca’s rational series for γ. Using a generalization of Vacca’s series to integer bases b ≥ 2, due in part to Ramanujan, we extend Addison’s cubic, rational, base 2 series for γ to faster base b series. Open problems on further extensions of the results are discussed, and a history of the formulas is given.

Keywords

Alternating Euler constant Acceleration of series Binary expansion Euler’s constant Generalized Euler constant Rational series Vacca’s series 

Notes

Acknowledgements

I am grateful to Stefan Krämer and Wadim Zudilin for valuable comments, and to Tanguy Rivoal for sending me a draft of [15].

References

  1. 1.
    Addison, A.W.: A series representation for Euler’s constant. Am. Math. Mon. 74, 823–824 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Allouche, J.-P., Shallit, J., Sondow, J.: Summation of series defined by counting blocks of digits, J. Number Theory 123, 133–143 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Barrow, D.F.: Solution to Problem 4353, Am. Math. Mon. 58, 117 (1951)MathSciNetGoogle Scholar
  4. 4.
    Behrmann, A.: Problem 5460, Am. Math. Mon. 74, 206 (1967)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Berndt, B.C., Bowman, D.C.: Ramanujan’s short unpublished manuscript on integrals and series related to Euler’s constant. In: Thera, M. (ed.) Constructive, Experimental and Nonlinear Analysis, pp. 19–27. American Mathematical Society, Providence (2000)Google Scholar
  6. 6.
    Carlitz, L.: Advanced Problem 5601, Am. Math. Mon. 75, 685 (1968)CrossRefGoogle Scholar
  7. 7.
    Franklin, F.: On an expression for Euler’s constant, J. Hopkins Circ. II, 143 (1883)Google Scholar
  8. 8.
    Gerst, L.: Some series for Euler’s constant, Am. Math. Mon. 76, 273–275 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Harborth, H.: Solution to Problem 5601, Am. Math. Mon. 76, 568 (1969)Google Scholar
  10. 10.
    Jacobsthal, E.: Ueber die Eulersche Konstante, Math.-Naturwiss. Blätter 9, 153–154 (1906)Google Scholar
  11. 11.
    Krämer, S.: Die Eulersche Konstante γ und verwandte Zahlen. Diplomarbeit, Mathematisches Institut der Georg-August-Universität Göttingen (2005)Google Scholar
  12. 12.
    Nielsen, N.: Een Raekke for Euler’s Konstant, Nyt. Tidss. Math. 88, 10–12 (1897)Google Scholar
  13. 13.
    Pilehrood, K.H., Pilehrood, T.H.: Arithmetical properties of some series with logarithmic coefficients, Math. Z. 255, 117–131 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Ramanujan, S.: The Lost Notebook and Other Unpublished Papers (Introduction by G.E. Andrews). Springer, Berlin; Narosa Publishing House, New Delhi (1988)Google Scholar
  15. 15.
    Rivoal, T.: Polynômes de type Legendre et approximations de la constante d’Euler (2005, unpublished notes); available at http://www-fourier.ujf-grenoble.fr/~rivoal/
  16. 16.
    Sandham, H.F.: Advanced Problem 4353, Am. Math. Mon. 56, 414 (1949)CrossRefGoogle Scholar
  17. 17.
    Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences (2008); published at http://www.research.att.com/~njas/sequences/
  18. 18.
    Sondow, J.: Double integrals for Euler’s constant and ln(4 ∕ π) and an analog of Hadjicostas’s formula, Am. Math. Mon. 112, 61–65 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Sondow, J.: New Vacca-type rational series for Euler’s constant and its “alternating” analog ln(4 ∕ π) (2005, preprint); available at http://arXiv.org/abs/math/0508042v1
  20. 20.
    Sondow, J., Hadjicostas, P.: The generalized-Euler-constant function γ(z) and a generalization of Somos’s quadratic recurrence constant, J. Math. Anal. Appl. 332, 292–314 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Sondow, J., Zudilin, W.: Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper, Ramanujan J. 12, 225–244 (2006); expanded version available at http://arXiv.org/abs/math/0304021
  22. 22.
    Vacca, G.: A new series for the Eulerian constant \(\gamma =.577\ldots \), Quart. J. Pure Appl. Math. 41, 363–364 (1910)zbMATHGoogle Scholar
  23. 23.
    van Lint, J. H.: Solution to Problem 5460, Am. Math. Mon. 75, 202 (1968)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.New YorkUSA

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