Is there a "finitary" solution to the Basel problem? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T23:08:11Z http://mathoverflow.net/feeds/question/9465 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/9465/is-there-a-finitary-solution-to-the-basel-problem Is there a "finitary" solution to the Basel problem? Qiaochu Yuan 2009-12-21T09:17:41Z 2009-12-21T19:23:59Z <p>Gabor Toth's <a href="http://www.amazon.com/Glimpses-Algebra-Geometry-Undergraduate-Mathematics/dp/0387982132" rel="nofollow">Glimpses of Algebra and Geometry</a> 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 $N(r)$ be the number of ordered pairs $(a, b) \in \mathbb{Z}^2$ such that $a^2 + b^2 \le r^2$, i.e. the number of lattice points in the ball of radius $r$. Then if $r_2(n)$ is the number of ordered pairs $(a, b) \in \mathbb{Z}^2$ such that $a^2 + b^2 = n$, it follows that $N(r^2) = 1 + r_2(1) + ... + r_2(r^2)$.</p> <p>On the other hand, once one has characterized the primes which are a sums of squares, it's not hard to show that $r_2(n) = 4(d_1(n) - d_3(n))$ where $d_i(n)$ is the number of divisors of $n$ congruent to $i \bmod 4$. So we want to count the number of divisors of numbers less than or equal to $r^2$ congruent to $i \bmod 4$ for $i = 1, 3$ and take the difference. This gives</p> <p>$\displaystyle \frac{N(r^2) - 1}{4} = \left\lfloor r^2 \right\rfloor - \left\lfloor \frac{r^2}{3} \right\rfloor + \left\lfloor \frac{r^2}{5} \right\rfloor \mp ...$</p> <p>and now the desired result follows by dividing by $r^2$ and taking the limit.</p> <p><strong>Question:</strong> Does a similar proof exist of the formula $\displaystyle \frac{\pi^2}{6} = 1 + \frac{1}{2^2} + \frac{1}{3^2} + ...$?</p> <p>By "similar" I mean one first establishes a finitary result with a clear number-theoretic or combinatorial meaning and then takes a limit.</p> http://mathoverflow.net/questions/9465/is-there-a-finitary-solution-to-the-basel-problem/9497#9497 Answer by Alon Amit for Is there a "finitary" solution to the Basel problem? Alon Amit 2009-12-21T18:49:16Z 2009-12-21T18:49:16Z <p>I think that the 14th and last proof in <a href="http://secamlocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf" rel="nofollow">Robin Chapman's collection</a> 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. </p>