Are there any formulas or identities that involves $\pi$ but have no obvious trigonometric interpretation?

Question

Every formula that involves $\pi$ has an underlying trigonometric interpretation but it is not usually obvious.

I wonder if there are any formula like the Gaussian integral $f_{n}(x)=\sqrt{\frac{1}{2\pi n^2}} \exp\left(\frac{-x^2}{2n^2}\right)$ which contains $\pi$ but does not have any obvious trigonometric interpretation.

Answer 1

The number of squarefree numbers (numbers not divisible by any square number other than 1) less than $n$ is asymptotically equal to $6n/\pi^2$.

The number of pairs $a,b$ with $0\le a\le n$, $0\le b\le n$, $\gcd(a,b)=1$ is asymptotically $6n^2/\pi^2$.

I wrote an expository paper on the number of three-term geometric progressions with all terms less than $n$, and the answer came out to involve $\pi$. I'll come back with more details.

EDIT: Here's the paper. The number of three-term geometric progressions $a,b,c$ with $1\le a\le b\le c\le n$ is asymptotically $(3/\pi^2)n\log n$.

Answer 2

The Basel problem was one of the earlier examples of this. $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}.$$