The Catalan Constant

Today I turned 25977 days old and this number is prime. One of its claims to fame is that it is a member of OEIS A104919:

A104919

Primes from merging of 5 successive digits in decimal expansion of Catalan's constant.

I was familiar with Catalan numbers and in fact posted about them in this post from the 30th of September 2015 and another post from the 15th of April 2018. These numbers, and the eponymous constant, are named in honour of the Belgium mathematician Eugène Catalan. So what is the Catalan constant? It is defined as follows:$$\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^3} = \frac{1}{1^3} - \frac{1}{3^3} + \frac{1}{5^3} - \frac{1}{7^3} + \frac{1}{9^3} - \cdots$$It is equal to 0.915965594177219015054603514932384110774 ... but the interesting thing is that no one knows whether it is irrational and, if it is, whether it is transcendental. This is odd because a similar looking sequence is known to be transcendental viz.:$$\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^3} = \frac{1}{1^3} - \frac{1}{3^3} + \frac{1}{5^3} - \frac{1}{7^3} + \frac{1}{9^3} - \cdots = \frac{\pi^3}{32}$$As of July 16th 2019, the Catalan constant is known to 600,000,000,100 decimal places. This site lists the constant to 300,000 decimal places with the introduction:

Catalan constant to 300000 digits computed on September 29, 1996 by using a Sun Ultra-Sparc in 1 day 8 hour 15 min 15 sec 55 hsec. The algorithm used is the standard series for Catalan, accelerated by an Euler transform. The algorithm was implemented using the LiDIA library for computational number theory and it is part of the multiprecision floating-point arithmetic of the package.

This number of decimal places was the record for 1996 and it was more than sufficient for me to generate the OEIS sequence A104919 up to and including 25997. I simply copied a few thousand of the first few digits into a list and applied the SageMath algorithm shown in this permalink.

Of course, the digits can be generated using a wide variety of formulae. One of the best is the following: $$\frac{\pi}{8}\log\left(2 + \sqrt{3}\right) + \frac{3}{8}\sum_{n=0}^\infty \frac{1}{(2n+1)^2 \binom{2n}{n}}$$I tried this but wasn't successful in generating more than the first two members of the sequence so I abandoned this approach and copied and pasted instead! I'm always fascinated by these types of sequences because of their similarity to the famous Riemann zeta function:$$\zeta(s) =\sum_{n=1}^\infty\frac{1}{n^s}$$In fact the summation that generates the Catalan constant is the Dirichlet beta function (also known as the Catalan beta function) and it is closely related to the Riemann zeta function. It is defined as:$$\beta(s) = \sum_{n=0}^\infty \frac{(-1)^n} {(2n+1)^s}$$When \(s=2\), \( \beta(2)\) gives the Catalan constant. You can read more here about the Dirichlet beta function.