TITLE: Are the unit partial quotients of $\pi, \log(2), \zeta(3) $ and other constants $all$ governed by $H=0.415\dots$?
QUESTION [9 upvotes]: Khinchin showed that given the simple continued fraction of a real number,
$$r = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1} {\ddots}}}$$
then it is almost always true that the partial quotients $a_i$ satisfy,
$$K = \lim_{n \rightarrow \infty } \left( a_1 a_2 ... a_n \right) ^{1/n} =2.685452\dots$$
where $K$ is Khinchin's constant. (Some exceptional $r$ are the rationals, roots of quadratic equations, and rational powers of $e$.)

Q1: Given $n$, let $T_n$ be the total number of partial quotients $a_i = 1$. Is it almost always true that,
  $$H=\lim_{n \rightarrow \infty } \frac{T_n}{n} = 0.415\dots$$
  exists and converges? If it does, can $H$ be expressed in terms of $K$ and other constants?

Numerical evidence for various transcendental and algebraic constants are given below with $n=10^k$ and entries as $T_n$:
$$\begin{array}{|c|c|c|c|c|}
\hline
\text{constant}&10^3&10^4&10^5&10^6&10^7\\
\hline
\pi&412& 4206& 41494& 414526& 4148280\\
\Gamma\big(\tfrac{1}{2}\big)&417& 4178& 41620& 415352& 4151849\\
\log(2)&433& 4148& 41430& 415443&-\\
\log(3)&429& 4170& 41458& 414919&-\\
T&396& 4084& 41172& 414458&-\\
P&410& 4087& 41364& 415180&-\\
K&418& 4111& 41379&-&-\\
C&412& 4147& 41543&-&-\\
\zeta(3)&418& 4223&-&-&-\\
\hline
\end{array}$$
where $T, P, K, C$ are the tribonacci, plastic, Khinchin, and Catalan constants.

Q2: Anybody able to fill in the blanks? Or extend it to $n>10^{7}$ so we can have more decimal digits of $H$? (I know the continued fraction for $\pi$ has been computed to more than $n>10^{10}$ terms.)

REPLY [15 votes]: A probability distribution of the continued fraction expansion terms follows the Gauss-Kuzmin distribution for almost all irrational numbers:
$$p(k)=-\log_2\left(1-\frac{1}{(k+1)^2}\right)=\log_2\frac{(k+1)^2}{k(k+2)}$$
All generalized Khinchin's constants (including $K=K_0$, the geometric mean), are derived from this distribution.
In this case, you seek the fraction of terms with value $1$, which is
$$p(1)=\log_2 \frac{4}{3}\approx 0.4150375$$
So it turns out this constant you observed is expressible with elementary functions.