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.

Twin Prime Constant

The following standard mathematical constants are defined in SageMath (link):

• 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. In this post, I'll be examining the former but first let's use SageMathCell to approximate the constants in the above list. See Figure 1.

Figure 1: permalink

Figure 2 zooms in a little closer on the output from Figure 1:

Figure 2

So what is the twin prime constant? Well, according to this source, the famous mathematical pair of Hardy and Littlewood conjectured that there are about:$$2  \prod_{p \geq 3} \frac{p(p-2)}{(p-1)^2} \int_2 ^x \frac{ \text{d}x}{(\log{x})^2}  \approx 1.320323632 \int_2 ^x \frac{ \text{d}x}{(\log{x})^2}$$twin primes less than or equal to \(x\) where the infinite product is the twin prime constant. In other words, the twin prime constant is given by$$\prod_{p \geq 3} \frac{p(p-2)}{(p-1)^2} \approx 0.66016181584686957393$$The  agreement between the calculated number of twin primes and the actual number gets better and better as \(x\) gets larger. Figure 3 shows the progression:

Figure 3

This is a complex topic and I won't go any deeper into here, as my intention was simply to explain what the constant represents and how it arises.

Euler–Mascheroni constant and the Meissel–Mertens constant

I've not written explicitly about either the Euler–Mascheroni constant or the Meissel–Mertens constant before, although the former is made mention of in a Numberphile video that I referenced in a post titled The Harmonic Series on October 12th 2016. 

Let's recount that the harmonic series is simply \(\zeta(1)\) and so:$$\zeta(1)=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+ \dots =\sum_{n=1}^{\infty}\frac{1}{n}$$While this sequence does diverge it does so very slowly and that's what my post The Harmonic Series was all about. The Euler-Mascheroni constant can be defined as:$$\begin{align}

\gamma &= \lim_{n\to\infty}\left(-\log n + \sum_{k=1}^n \frac1{k}\right)\\

&=\int_1^\infty\left(-\frac1x+\frac1{\lfloor x\rfloor}\right)\,dx.

\end{align}$$Here, \(\lfloor x\rfloor\) represents the floor function. The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is:

0.57721566490153286060651209008240243104215933593992... 

Below I've embedded the Numberphile video referred to earlier as it's really quite informative.

Like the harmonic series, the sum of the reciprocals of the prime numbers diverges also and even more slowly. The Meissel-Mertens constant is defined as:$$M = \lim_{n \rightarrow \infty } \left( \sum_{p \leq n} \frac{1}{p} - \ln(\ln n) \right)=\gamma + \sum_{p} \left[ \ln\! \left( 1 - \frac{1}{p} \right) + \frac{1}{p} \right]$$where \( \gamma \) is the Euler-Mascheroni constant. The value of M is approximately:

M ≈ 0.2614972128476427837554268386086958590516... 

Figure 1: source

The two constants are thus intimately linked. It's easy to generate approximations of these functions using SageMathCell. See Figure 2.

Figure 2: permalink

Looking at the results in Figure 2, it can be seen that:

Approximation of Euler-Mascheroni constant up to 100000 is 0.577220664893197

Approximation of Miessel-Mertens constant up to 100000 is 0.261801821365208

The light grey digits do not correspond to the known digits for these constants. It can be seen that the approximation to the Miessel-Mertens constant is less accurate than for the Euler-Mascheroni constant, reflecting the log(log) computation for the former versus the log computation for the latter.

For a post that shows how to determine the sum of the alternating harmonic series, see my post titled Alternating Series Test from April 23rd 2021. The alternating harmonic series converges thus:$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4} \dots=\ln(2) \approx 0.693147180559945 \dots$$See also:

• Biography of Ernet Meissel
• Biography of Franz Mertens
• Biography of Lorenzo Mascheroni
• Biography of Leonhard Euler

Feigenbaum Constants

The Feigenbaum constants are right up there with the big ones but it was only after watching a Numberphile video that I understood the significance of the delta 𝛅 constant. Here is the video that I watched:

The Feigenbaum constant 𝛿 is defined by Wikipedia as follows:

It's easier to see with a specific example of \( \text{f}(x) \), an old friend from Chaos Theory namely \(a  x(1-x) \) and a table of values for \(a\):

I created a spreadsheet in Google Sheets to illustrate the bifurcation process. Below is an example of the first bifurcation when a=3.3:

Below is an example of the second bifurcation, taken when a=3.5:

And so on it goes.