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I realized that I know of several ways how to prove that $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$, but I have no idea why I would want to know the answer in the first place.

Answers I have found by myself:

  • the probability of two integers chosen at random to be prime to each other is $\frac{6}{\pi^2}$. The proof is understandable by a smart undergraduate. This is the kind of motivation I am looking for.
  • by the inverse square law of sound, a line of cars blowing their horns at a car stopped in a one lane road will sound $\frac{\pi^2}6$ louder than a single car. Or similarly, intensity of traffic lights on a long road at night,...
  • destructive testing of $n$ wooden beams will break on average $H_n=1+\frac12+\frac13+\ldots+\frac1n$ beams with a variance of $H_n-\sum_{k=1}^{n}\frac1{k^2}$.
  • $\zeta(2)=\frac{\pi^2}{6}$ is all over quantum mechanics. The simplest, most understandable example I have found is Johnson-Nyquist noise. Still, there is a lot of physics background required to figure it out, so this does not satisfy me much.

Are there any other good reason why to compute $\zeta(2)$ ? Why were Mengoli, Euler and other mathematicians from the Enlightenment interested in the answer ? Any application in physics, chemistry, economics,... ? As far as I am concerned, the closer to reality, the better.

Thanks in advance for any help.

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  • $\begingroup$ I don't know the history but I suspect Euler and others wanted to know the value of $\zeta(2)$ out of mere curiosity - they knew the series converged. Euler knew the relationship between $\zeta$ and the primes. Of course not the quantum mechanics. $\endgroup$ – Ethan Bolker Aug 1 '18 at 11:35
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    $\begingroup$ After the geometric series and the Leibniz series for ${\pi\over2}$ and $\log 2$ this is the next simplest series you could think of. It is natural to come up with this problem. $\endgroup$ – Christian Blatter Aug 5 '18 at 13:06
  • $\begingroup$ @ChristianBlatter I believe the Leibniz series you are thinking of is $\pi/4$ and that came after the Basel problem. Mengoli should have been motivated by the Wallis product for $\pi/2$ which he showed was correct in 1652 ? $\endgroup$ – Antonio Hernandez Maquivar Jan 23 at 16:45
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I know two other problem involving this identity.

$1.$ The probability of integers chosen at random to be square-free is $\frac{6}{\pi^2}$. The proof is similar to the first problem you mentioned.

$2.$ Parisi conjecture, which is not a conjecture anymore. It's about finding minimum weighted matching in bipartite graphs. You may find a script about that here.

But most of all I recommend to read this article by Raymond Ayoub about Euler and $\zeta$-function.

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