A new continued fraction for Euler’s constant

Aldo Peretti
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 3, 1997, Number 1, Pages 23—34
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Aldo Peretti
Facultad de Ciencia y Tecnologia – Universidad del Salvador
Rodriguez Peña 640
(1020) Buenos Aires, Argentina

Abstract

A new continued fraction is obtained for Euler constant C, namely:

    \[C = \frac{1}{2} - \frac{1}{3} = \Bigg \{ \frac{ \Big ( \frac{2r - 1}{r-1} \Big )^2 \mid }{ \mid \frac{-2^r  + r}{r(r+1)} } + \frac{2^r + r \mid}{ \mid r^2} +  r^2 \Big \{ \frac{(2^r + m)^2 \mid }{ \mid 1}  \Big \}_{m=1}^{m=2^r - 1}  \Bigg \}_{r = 2}^{r = \infty} \]

It is considered the possibility to prove the irrationality and transcendency of the constant by means of this expansion.

References

  1. R. Ayoub: “Partial triumph or total failure?”. Math. Intelligencer, vol. 7, N° 2 (1985) p. 55-58.
  2. D. Bailey: “Numerical results on the transcendence of constants involving \pi, e and Euler’s constant”. Math. Comp. vol. 50, (1988) p. 275-282
  3. W. A. Beyer and M. S. Waterman: “Error analysis of a computation of Euler’s constant”. Math. Comp. vol. 28 (1974) p. 599-604
  4. W. A. Beyer and M. S. Waterman: “Decimals and partial quotients of Euler’s constant and Ln2”. Math. Comp. vol. 28 (1974) p. 667
  5. R. P. Brent: “Computation of the regular continued fraction for Euler’s constant”. Math. Comp. vol. 31 (1977) p. 771-777
  6. R. P. Brent and E. Me. Millan: “Some new algorithms for high precision computation of Euler’s constant”. Math. Comp. vol. 34 (1980) p. 305-312
  7. A. Froda: “La constante d’Euler est irrationelle”. Atti. Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Natur. (8) 38 (1965) p. 338-344
  8. A. Froda: “Nouveaux criteres parametriques d’irrationalite”. C. R. Acad. Sci. Paris 261 (1965) p. 3012-3015
  9. J. W. Glaisher: “History of Euler’s constant”. Messenger of Math. Vol. 1 (1872) p. 25- 30
  10. H. W. Gould: “Some formulas for Euler’s constant”. Bull. Numb. Theory. Vol. X (1986) N° 2, p. 2-9
  11. Hermite-Stieltjes: Correspondance. Lettre 216, p.459, Gauthier Villars, Paris (1905).
  12. S. Selberg: “Bemerkung zu einer arbeit von Viggo Bran über die Riemannsche zetafunktion”. Norske Vid. Selsk. Forh. XIII N° 5 (1940), p. 17-19
  13. D. W. Sweeney: “On the computation of Euler’s constant”. Math. Comp. 17 (1963), p. 170-178
  14. O. Perron: “Die Lehre von Kettenbriichen”. 2nd. Edition, 1929. Teubner, Leipzig
  15. C. Brezinski: “History of Continued Fractions and Pade Approximants”. Springer Verlag. 1991
  16. G. Carr: “Formulas and theorems in Mathematics”. Chelsea edition.

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Cite this paper

Peretti, A. (1997). A new continued fraction for Euler’s constant. Notes on Number Theory and Discrete Mathematics, 3(1), 23-34.

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