Lochs's constant

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Number

$L$

0.9702701143920339257402560192100108337812847047851612866103505299312541998917370480362126749080290265

Definition

Lochs's theorem [1] says that for almost every real number the first $n$ decimal digits determine, asymptotically, $Ln$ terms of the regular continued fraction expansion [2], where $L=\frac{6\log 2\log 10}{\pi^2}$. This table holds $L$.

Formulas

(1)

$L=\frac{6\log 2\log 10}{\pi^2}=\frac{\log 10}{2\beta}$, where $\pi$ is $\pi$.

Comments

(2)

Writing $\beta=\frac{\pi^2}{12\log 2}$ for the Khinchin-Lévy constant $\beta$, the constant is $L=\frac{\log 10}{2\beta}$: a decimal digit is worth $L$ partial quotients because the denominators of the convergents grow like $e^{\beta n}$.

(4)

MathWorld records a second, nonstandard use of the name for a constant related to Porter's constant [3]. This table holds the constant of Lochs's theorem, which is the usual one.

Programs

(P1)

Sage

R = RealBallField(400)
6 * R(2).log() * R(10).log() / R.pi()^2

References

[1]

G. Lochs, Vergleich der Genauigkeit von Dezimalbruch und Kettenbruch, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 27 (1964), 142-144. (doi) (MR)

Links

[2]

MathWorld: Regular Continued Fraction

[3]

MathWorld: Lochs' Constant

[4]

OEIS A086819: decimal expansion of Lochs's constant

Similar tables

Lévy's constant — $L=\frac{\log 10}{2\beta}$, so this constant is that one read as a rate of conversion between decimal digits and partial quotients

Khinchin's means $K_p$ — the almost-sure limits of the power means of the partial quotients of the same expansion

$\pi$ — appears in the closed form (1)

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: yes

How they were obtained:

The value follows from (1) and was computed in real ball arithmetic, with $\pi$, $\log 2$ and $\log 10$ taken as balls rather than as rounded constants, so every stored digit is covered by the enclosure.

It was compared with the decimal expansion in OEIS A086819 [4], and the identity $L=\frac{\log 10}{2\beta}$ was checked against the Khinchin-Lévy constant in Lévy's constant.