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Q: Is there a "finitary" solution to the Basel problem?

Qiaochu YuanGabor Toth's Glimpses of Algebra and Geometry contains the following beautiful proof (perhaps I should say "interpretation") of the formula $\displaystyle \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} \mp ...$, which I don't think I've ever seen before. Given a non-negative integer $r$, let $N(r...

A: Is there a "finitary" solution to the Basel problem?

Alon AmitI think that the 14th and last proof in Robin Chapman's collection is just that. It relies on the formula for the number of representations of an integer as a sum of four squares, which is kind of overkill, but anyway.

What a fantastic proof, nonetheless! — Harrison Brown Dec 21, 2009 at 22:11
The link is not opening for me; I suspect it has been migrated to: empslocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdfBenjamin Dickman yesterday
Thank you. Fixed the link. — Alon Amit 40 mins ago
Since Alon Amit's edit bumped that question, I have edited a dead link in another answer: mathoverflow.net/posts/386574/revisions
A: Is there a "finitary" solution to the Basel problem?

Ian AgolA somewhat different perspective to the Basel problem relates $\zeta(2)$ to the volume of $SL_2(\mathbb{R})/SL_2(\mathbb{Z})=\zeta(2)/2$. They compute this volume via a count of lattice points. One can also compute this via Gauss-Bonnet as a circle bundle over the modular curve $\mathbb{H}^2/PSL_...


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