Lochs' Constant

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 ,

			(1)
			(2)
			(3)

(OEIS A086819; Lochs 1964). This number is sometimes known as Lochs' constant.

The reciprocal of this constant is

			(4)
			(5)

(OEIS A062542; Finch 2003, p. 60).

Lochs' constant is related to the Lévy constant  by

			(6)
			(7)

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."