The Euler-Mascheroni constant is denoted (or sometimes ) and has the numerical value

(1) |

It is not known if this constant is Irrational, let alone Transcendental. However, Conway and Guy (1996) are ``prepared to bet that it is transcendental,'' although they do not expect a proof to be achieved within their lifetimes.

The Euler-Mascheroni constant arises in many integrals

(2) | |||

(3) | |||

(4) |

and sums

(5) | |||

(6) | |||

(7) | |||

(8) | |||

(9) |

where is the Riemann Zeta Function and are the Bernoulli Numbers. It is also given by the Euler Product

(10) |

(11) |

Infinite Products involving also arise from the *G*-Function
with Positive Integer . The cases and give

(12) | |||

(13) |

The Euler-Mascheroni constant is also given by the limits

(14) | |||

(15) | |||

(16) |

(Le Lionnais 1983).

The difference between the th convergent in (6) and is given by

(17) |

(18) |

(19) |

(20) |

(21) |

(22) |

The symbol is sometimes also used for

(23) |

Odena (1982-1983) gave the strange approximation

(24) |

(25) | |||

(26) | |||

(27) | |||

(28) |

No quadratically converging algorithm for computing is known (Bailey 1988). 7,000,000 digits of have been computed as of Feb. 1998 (Plouffe).

**References**

Bailey, D. H. ``Numerical Results on the Transcendence of Constants Involving , , and Euler's Constant.''
*Math. Comput.* **50**, 275-281, 1988.

Beeler, M.; Gosper, R. W.; and Schroeppel, R. *HAKMEM.* Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

Brent, R. P. ``Computation of the Regular Continued Fraction for Euler's Constant.'' *Math. Comput.* **31**, 771-777, 1977.

Brent, R. P. and McMillan, E. M. ``Some New Algorithms for High-Precision Computation of Euler's Constant.''
*Math. Comput.* **34**, 305-312, 1980.

Castellanos, D. ``The Ubiquitous Pi. Part I.'' *Math. Mag.* **61**, 67-98, 1988.

Conway, J. H. and Guy, R. K. ``The Euler-Mascheroni Number.'' In *The Book of Numbers.* New York: Springer-Verlag,
pp. 260-261, 1996.

de la Vallée Poussin, C.-J. Untitled communication. *Annales de la Soc. Sci. Bruxelles* **22**, 84-90, 1898.

DeTemple, D. W. ``A Quicker Convergence to Euler's Constant.'' *Amer. Math. Monthly* **100**, 468-470, 1993.

Dirichlet, G. L. *J. für Math.* **18**, 273, 1838.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/euler/euler.html

Flajolet, P. and Vardi, I. ``Zeta Function Expansions of Classical Constants.'' Unpublished manuscript, 1996. http://pauillac.inria.fr/algo/flajolet/Publications/landau.ps.

Gerst, I. ``Some Series for Euler's Constant.'' *Amer. Math. Monthly* **76**, 273-275, 1969.

Glaisher, J. W. L. ``On the History of Euler's Constant.'' *Messenger of Math.* **1**, 25-30, 1872.

Gradshteyn, I. S. and Ryzhik, I. M. *Tables of Integrals, Series, and Products, 5th ed.* San Diego, CA:
Academic Press, 1979.

Knuth, D. E. ``Euler's Constant to 1271 Places.'' *Math. Comput.* **16**, 275-281, 1962.

Le Lionnais, F. *Les nombres remarquables.* Paris: Hermann, p. 28, 1983.

Plouffe, S. ``Plouffe's Inverter: Table of Current Records for the Computation of Constants.'' http://www.lacim.uqam.ca/pi/records.html.

Sloane, N. J. A. Sequences A033091, A033092, A033149, A046114, A046114, A001620/M3755, and A002852/M0097 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Sweeney, D. W. ``On the Computation of Euler's Constant.'' *Math. Comput.* **17**, 170-178, 1963.

Vacca, G. ``A New Series for the Eulerian Constant.'' *Quart. J. Pure Appl. Math.* **41**, 363-368, 1910.

Young, R. M. ``Euler's Constant.'' *Math. Gaz.* **75**, 187-190, 1991.

© 1996-9

1999-05-25