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Draw line segments from the centre of a unit circle to two uniformly random points on the circle, forming two regions of area $A_1$ and $A_2$.

enter image description here

It is easy to show that the expectation of the product of the areas $E(A_1A_2)=\frac{1}{\pi}\int_0^\pi\frac{1}{2}x\left(\pi-\frac{1}{2}x\right)dx=\frac{\pi^2}{6}$, which happens to be the answer to the Basel problem, $\sum\limits_{n=1}^\infty \frac{1}{n^2}$.

Is this just a coincidence? Or can we solve the Basel problem by showing that $\sum\limits_{n=1}^\infty \frac{1}{n^2}=E(A_1A_2)$, and only then calculating $E(A_1A_2)=\frac{\pi^2}{6}$?

The reason I think it might not be a coincidence, is that $E(A_1A_2)$ and $\sum\limits_{n=1}^\infty \frac{1}{n^2}$ are both simple, non-arbitrary constructions. It's not like we're choosing special constants just to make two things equal.

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    $\begingroup$ I believe 3blue1brown presented a geometric proof of the Basel problem (presumably not originally his, but that's the only reference I have off the top of my head) so it may be worth checking that out $\endgroup$
    – FShrike
    Commented Dec 12, 2022 at 14:07
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    $\begingroup$ My guess is that it is a coincidence. $\endgroup$
    – Peter
    Commented Dec 12, 2022 at 14:16
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    $\begingroup$ @FShrike I guess you're referring to this. Great video, but I don't think it's the same as what I'm thinking of here (at least not obviously). $\endgroup$
    – Dan
    Commented Dec 12, 2022 at 14:19
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    $\begingroup$ Why is the product of areas an interesting quantity to consider ? If there was a geometrical proof for the sum of inverse squares related to this question, I would expect it to involve something more fundamental about geometry than a product of areas of sections in a disk. A product of areas seems arbitrary not knowing the problem that you are considering. $\endgroup$
    – userrandrand
    Commented Dec 12, 2022 at 14:40
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    $\begingroup$ @userrandrand I think product of areas in a circle can be quite interesting, for example: question 1, question 2, question 3. $\endgroup$
    – Dan
    Commented Dec 12, 2022 at 14:51

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