nt.number theory - Is there a "finitary" solution to the Basel problem?

Gabor 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 $N(r)$ 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 $N(r^2) = 1 + r_2(1) + ... + r_2(r^2)$.

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

$displaystyle frac{N(r^2) - 1}{4} = leftlfloor r^2 rightrfloor - leftlfloor frac{r^2}{3} rightrfloor + leftlfloor frac{r^2}{5} rightrfloor mp ...$

and now the desired result follows by dividing by $r^2$ and taking the limit.

Question: Does a similar proof exist of the formula $displaystyle frac{pi^2}{6} = 1 + frac{1}{2^2} + frac{1}{3^2} + ...$?

By "similar" I mean one first establishes a finitary result with a clear number-theoretic or combinatorial meaning and then takes a limit.