Khinchin's Constant

As I said in my previous post, Twin Prime Constant, the following standard mathematical constants are defined in SageMath:

• pi
• golden_ratio
• log2
• euler_gamma
• catalan
• khinchin
• twinprime
• mertens

I'm familiar with all the constants above except two: the twinprime constant and the khinchin constant. I've dealt with the former and so this post is about the latter. To quote from Wikipedia:

Aleksandr Yakovlevich Khinchin proved that for almost all real numbers \(x\), coefficients \(a_i\) of the continued fraction expansion of \(x\) have a finite geometric mean that is independent of the value of \(x\) and is known as ''Khinchin's constant''. That is, for$$x = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\ddots}}}}$$it is almost always true that$$\lim_{n \rightarrow \infty } \left( a_1 a_2 ... a_n \right) ^{1/n} = K_0$$where \(K_0\) is Khinchin's constant which is equal to$$\prod_{r=1}^\infty {\left( 1+{1\over r(r+2)}\right)}^{\log_2 r}  \approx 2.6854520010\dots$$Although almost all numbers satisfy this property, it has not been proven for ''any'' real number ''not'' specifically constructed for the purpose. Among the numbers \(x\) whose continued fraction expansions are known ''not'' to have this property are rational numbers, roots of quadratic equations (including the golden ratio, the square roots of integers) and the base of the natural logarithm \(e\).

\( \pi \), the Euler–Mascheroni constant \( \gamma \), and Khinchin's constant itself, based on numerical evidence, are thought to be among the numbers whose geometric mean of the coefficients \(a_i\) in their continued fraction expansion tends to Khinchin's constant. However, none of these limits have been rigorously established. It is not known whether Khinchin's constant is a rational, algebraic irrational or transcendental number.

Let's look at Wolfram MathWorld's article on the Euler-Mascheroni Constant Continued Fraction in which the continued fraction for \( \gamma \) is given as:

[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...] (OEIS A002852)

SageMathCell (permalink) can display what this continued fraction looks like. See Figure 1.

Figure 1: permalink

Figure 2 shows a plot of the progressive harmonic means of successive values of \(a_1^{1/1}, (a_1.a_2)^{1/2}, (a_1.a_2...a_n)^{1/n} \) that appear to approach Khinchin's constant, although this has not been rigorously proven:

Figure 2: source

Figure 2 shows a rather more complicated plot of the values \( (a_1.a_2,...,a_n)^{1/n} \) for \(n\)=1 to 500 and \(x=\pi, \sin 1\), the Euler-Mascheroni constant \( \gamma\), and the Copeland-Erdős constant \(C\). The horizontal line marked \(K\) in the plot is Khinchin's constant.

Figure 2: source

I've mentioned the geometric mean briefly in posts titled Root-Mean-Square And Other Means on September 13th 2020 and Reciprocals of Primes on October 30th 2021. This mean is one of the three Pythagorean means along with the arithmetic and harmonic. I definitely need to dedicate a post to these three means.