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I know that the equality $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$ can be proved in numerous ways by using the Fourier series. However, is there a way to derive it using more fundamental tools? I've tried: $$ \sum_{n=1}^\infty \frac{1}{n^2} = \sum_{n=1}^\infty \int_0^1 x^{n-1} dx \int_0^1 y^{n-1} dy = \int_0^1 dx\int_0^1 dy \frac{1}{1-xy}$$ and by changing variables I was able to write it in several other forms:$$ = -\int_0^1\frac{\ln(1-x)}{x} dx = \int_0^\infty \frac{\ln (1+t)}{t(1+t)} dt = \int_0^\infty \frac{u }{e^u -1}du $$ but that's as far as I could get.

I specifically don't want to use Fourier series. More fundamental complex analysis, like contour integration, is fine.

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Yes, this can be done. Actually, that's what Tom Apostol did in an article he published in 1983; you can read the proof here (it's the first proof). He used a change of variable ($(x,y)\mapsto(x+y,x-y)$) in order to compute the integral$$\iint_{[0,1]\times[0,1]}\frac1{1-xy}\,\mathrm dx\,\mathrm dy.$$

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