The Basel Problem and The Variance of A Particle in a Box

Question

As a beginner of quantum mechanics, I have just learnt how to solve the wave function for a particle in a one-dimensional box. Letting the width of the one-dimensional box be $L$, the variance in position $\langle x^2\rangle$ regarding to the ground state is: $$ \langle x^2\rangle=\frac{L^2}{12}\left(1-\frac6{\pi^2}\right) $$ On the other hand, there is the Basel problem: $$ \frac{\pi^2}6=\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\cdots+\frac1{n^2}+\cdots $$ I had found this video on YouTube by 3blue1brown before and I found it fascinating how he made an intuitional analogy of the problem and the net-power-of-light-sources problem.
And I thought, wouldn't it be nice if there is also an analogy between the particle-in-a-box problem and the Basel problem, since $\dfrac{\pi^2}6$ appears (inversed) in the equation for $\langle x^2\rangle$? Can anyone come up with a good explanation for this correlation?
Or, maybe it is surely a mere coincidence. If you think so, since I am a sophomore, could you provide a hint for a better understanding of 'variance' regarding to this particle problem? If this particle problem and the Basel problem has nothing to do with each other, I guess the significance of $\pi$ has to do something with oscillation of the wave function, but I cannot find a better qualitative expression by myself.
This post is purely for fun and to nourish my naïve intuition, but I hope I could hear your ideas more than merely solving integral equations.