Probability of Two Random Integers having no Common Divisor

Theorem
Let $a$ and $b$ be integers chosen at random.

The probability that $a$ and $b$ are coprime is given by:
 * $P \left({a \perp b}\right) = \dfrac 1 {\zeta \left({2}\right)} = \dfrac 6 {\pi^2}$

where $\zeta$ denotes the zeta function.

The decimal expansion of $\dfrac 1 {\zeta \left({2}\right)}$ starts:
 * $\dfrac 1 {\zeta \left({2}\right)} = 0 \cdotp 60792 \, 71018 \, 54026 \, 6 \ldots$