“Liouville-Roth Constant”的概念、定义、习题解题方法及翻译-数学辞典

单词

Liouville-Roth Constant

释义

Liouville-Roth Constant

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

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

(1)

has (at most) finitely many solutions

for

and

Integers. Then the Liouville-Rothconstant (or Irrationality Measure) is defined as the threshold at which Liouville's Rational Approximation Theorem kicks in and

is no longer approximable by RationalNumbers,

(2)

There are three regimes:

(3)

The best known upper bounds for common constants are

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)

where

is Liouville's Constant,

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

See also Liouville's Rational Approximation Theorem, Roth's Theorem, Thue-Siegel-Roth Theorem

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.

随便看

数学辞典收录了8975条数学词条，基本涵盖了常用数学知识及数学英语单词词组的翻译及用法，是数学学习的有利工具。