The Basel Problem and Beyond

The Basel Problem asks for the exact sum, in closed form, of the summation:$$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2} + \cdots$$Leonhard Euler proved that the sum was \( \dfrac{\pi}{6}\) and the details of Euler's proof and other proofs can be found here. Euler of course went further and generalised his investigation to include:$$\zeta(s) =\sum_{n=1}^\infty\frac{1}{n^s}$$ This is the Riemann zeta function or Euler–Riemann zeta function \(\zeta(s)\) which is a function of the complex variable \(s\). In this post, I'll focus on the case where \(s=3\) and look at the exact value of the Riemann zeta function for this value:$$\begin{align}\zeta(3) &= \sum_{n=1}^\infty\frac{1}{n^3} \\&= \lim_{n \to \infty}\left(\frac{1}{1^3} + \frac{1}{2^3} + \cdots + \frac{1}{n^3}\right)\end{align}$$The value of this summation is called Apéry's constant. \(\zeta(3)\) was so named because of the French mathematician, Roger Apéry, who proved in 1978 that it is an irrational number. Here is an interesting Numberphile video about this mathematician's proof:

Up to the present time however, Apéry's constant has not been proven to be transcendental. There are many interesting ways of calculating the value of this constant. Two are shown below and taken from the Wikipedia article about the constant:$$\zeta(3) =\frac{1}{2}\int_0^\infty \frac{x^2}{e^x-1}\, dx $$

$$\zeta(3) =\frac{2}{3}\int_0^\infty \frac{x^2}{e^x+1}\, dx $$