# Probability of Random Integer being Square-Free

## Contents

## 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

- Probability of Two Random Integers having no Common Divisor, which is the same probability as this

## 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

- 1983: François Le Lionnais and Jean Brette:
*Les Nombres Remarquables*... (previous) ... (next): $0,60792 71018 \ldots$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $0 \cdotp 607 \, 927 \, 101 \ldots$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $0 \cdotp 60792 \, 7101 \ldots$

- Weisstein, Eric W. "Relatively Prime." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/RelativelyPrime.html