Apéry's constant
From specialfunctionswiki
Apéry's constant is the number $\zeta(3)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k^3}$, where $\zeta$ denotes the Riemann zeta function. This constant is notable because it is known in general that for integers $n$, $\zeta(2n)$ is a rational multiple of $\pi$ but no "nice" closed formula exists for values $\zeta(2n+1)$. Hence it became a notorious open problem to find $\zeta$ at odd integers.
Properties
Theorem: The following formula holds: $$\displaystyle\int_0^{\frac{\pi}{2}} x \log(\sin(x)) dx = \dfrac{7}{16}\zeta(3) - \dfrac{\pi^2}{8} \log(2).$$
Proof: █
Theorem: (Apéry) The number $\zeta(3)$ is irrational.
Proof: █
