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Date: 20-11-2019
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Date: 16-11-2020
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Date: 26-8-2020
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For a real number , let be the number of terms in the convergent to a regular continued fraction that are required to represent decimal places of . Then Lochs' theorem states that for almost all ,
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(OEIS A086819; Lochs 1964). This number is sometimes known as Lochs' constant.
The reciprocal of this constant is
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(OEIS A062542; Finch 2003, p. 60).
Lochs' constant is related to the Lévy constant by
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In the index and table of constants Finch (2003, pp. 546 and 596) refers to the quantity
(8) |
related to Porter's constant as "Lochs' constant," though this terminology appears to be nonstandard.
REFERENCES:
Bosma, W.; Dajani, K.; and Kraaikamp, C. "Entropy and Counting Correct Digits." Univ. Nijmegen Math. Report 9925, 1999.
Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.
Kintchine, A. "Zur metrischen Kettenbruchtheorie." Compos. Math. 3, 276-285, 1936.
Kraaikamp, C. "A New Class of Continued Fraction Expansions." Acta Arith. 57, 1-39, 1991.
Lévy, P. "Sur le developpement en fraction continue d'un nombre choisi au hasard." Compos. Math. 3, 286-303, 1936.
Lochs, G. "Vergleich der Genauigkeit von Dezimalbruch und Kettenbruch." Abh. Hamburg Univ. Math. Sem. 27, 142-144, 1964.
Perron, O. Die Lehre von den Kettenbrüchen, 3. verb. und erweiterte Aufl. Stuttgart, Germany: Teubner, 1954-57.
Sloane, N. J. A. Sequences A062542 and A086819 in "The On-Line Encyclopedia of Integer Sequences."
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