Why does $\pi$ appear in the probability of a number being square-free? The probability of a number being square-free (i.e., the number has no divisor that is a square, cube, etc.) is $6/\pi^2$. I have seen many appearances of $\pi$, and this is also similar to them. All of them can be explained intuitively why pi appears in them. But I don't see any connection between square-free numbers and $\pi$. So my question is:

Why $\pi$ appears here?

Note: I don't want rigorous proofs, I want intuitive explanations.

The most intuitive thing I can think of would be the Riemann Zeta function. You are working with squares, and you will be adding a bunch of squares. In a similar way, if you wanted cube free numbers, you would get Apéry's constant.
