*N.B. A detailed on-line essay by S. Finch
was the starting point for this entry.*

Let be a Real Number, and let be the Set of Positive Real Numbers
for which

(1) |

(2) |

(3) |

(4) | |||

(5) | |||

(6) | |||

(7) | |||

(8) | |||

(9) |

where is Liouville's Constant, is Apéry's Constant, and the lower bounds are 2 for the inequalities.

**References**

Borwein, J. M. and Borwein, P. B.
*Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.* New York: Wiley, 1987.

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

Hardy, G. H. and Wright, E. M. *An Introduction to the Theory of Numbers, 5th ed.*
Oxford: Clarendon Press, 1979.

Hata, M. ``Improvement in the Irrationality Measures of and .''
*Proc. Japan. Acad. Ser. A Math. Sci.* **68**, 283-286, 1992.

Hata, M. ``Rational Approximations to and Some Other Numbers.'' *Acta Arith.* **63** 335-349, 1993.

Hata, M. ``A Note on Beuker's Integral.'' *J. Austral. Math. Soc.* **58**, 143-153, 1995.

Stark, H. M. *An Introduction to Number Theory.* Cambridge, MA: MIT Press, 1978.

© 1996-9

1999-05-25