Probability of Three Random Integers having no Common Divisor

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Theorem

Let $a, b$ and $c$ be integers chosen at random.


The probability that $a, b$ and $c$ have no common divisor:

$P \left({\perp \left({a, b, c}\right)}\right) = \dfrac 1 {\zeta \left({3}\right)}$

where $\zeta$ denotes the zeta function:

$\zeta \left({3}\right) = \dfrac 1 {1^3} + \dfrac 1 {2^3} + \dfrac 1 {3^3} + \dfrac 1 {4^3} + \cdots$


The decimal expansion of $\dfrac 1 {\zeta \left({3}\right)}$ starts:

$\dfrac 1 {\zeta \left({3}\right)} = 0 \cdotp 83190 \, 73725 \, 80707 \ldots$

This sequence is A088453 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof


Sources