Probability of Random Integer being Square-Free

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Theorem

Let $a$ be an integer chosen at random.


The probability that $a$ is square-free is given by:

$\map \Pr {\neg \exists b \in \Z: b^2 \divides a} = \dfrac 1 {\map \zeta 2} = \dfrac 6 {\pi^2}$

where $\zeta$ denotes the zeta function.


The decimal expansion of $\dfrac 1 {\map \zeta 2}$ starts:

$\dfrac 1 {\map \zeta 2} = 0 \cdotp 60792 \, 71018 \, 54026 \, 6 \ldots$

This sequence is A059956 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Let $a$ be an integer chosen at random.

For $a$ to be square-free, it is necessary and sufficient that for all prime numbers $p$, it is not the case that $p^2$ is a divisor of $a$.

The probability that any particular integer is divisible by $p^2$ is $\dfrac 1 {p^2}$.

The probability that $a$ is not divisible by $p^2$ is therefore $1 - \dfrac 1 {p^2}$.


Whether or not $a$ is divisible by $p^2$ or divisible by $q^2$ for another prime number $q$ is independent of both $p$ and $q$.

Thus by the Product Rule for Probabilities, the probability that $a$ is not divisible by either $p^2$ or $q^2$ is $\paren {1 - \dfrac 1 {p^2} } \paren {1 - \dfrac 1 {q^2} }$.

This independence extends to all prime numbers.

That is, the probability that $a$ is not divisible by the square of any prime number is equal to the product of $1 - \dfrac 1 {p^2}$ over all prime numbers:

$\map \Pr {\neg \exists b \in \Z: b^2 \divides a} = \displaystyle \prod_{\text {$p$ prime} } \paren {1 - \dfrac 1 {p^2} }$

From Sum of Reciprocals of Powers as Euler Product:

$\displaystyle \map \zeta s = \prod_p \frac 1 {1 - p^{-s} }$

from which:

$\displaystyle \dfrac 1 {\map \zeta 2} = \prod_{\text {$p$ prime} } \paren {1 - \dfrac 1 {p^2} }$

where $\map \zeta 2$ is the Riemann $\zeta$ (zeta) function evaluated at $2$.

The result follows from Riemann Zeta Function of 2.

$\blacksquare$


Also see


Historical Note

According to François Le Lionnais and Jean Brette in their Les Nombres Remarquables of $1983$, this result is attributed to Ernesto Cesàro and James Joseph Sylvester in $1883$.

However, this has not been corroborated.


Sources