I blathered
This differs from my previous strange identities in that the (rapidly convergent) infinite product is squared. As s -> 1 (spacefilling), the sum converges extremely slowly. In the spacefilling (s=1) limit, this formula seems to ascribe half the spacefilled area to the interior. In other words, it gives an area midway between the s=0 m-gon and the s=1 2m-gram obtained by erecting isosceles right triangles on the sides of the m-gon.
Actually, the s=1 case seems only to oscillate about this midpoint. For m=2, summing from -2^n to 2^n, we approach pi^3/2 - 1.7072613 for odd n and pi^3/2 + 1.7072613 for even n. This is peculiar. The Fourier series Sum c_j e^(i j t) converges for all real t to the spacefilling function, yet something akin to an integral of this function fails to converge. --rwg