The calculations in my previous message approximate pi/2 (and not 1.61... as I wrongly stated), so this part of the problem is solved, but how do the primes get in? Euler's infinite product is a genuine link of pi and the primes, for all primes appear as numerators, while the denominators are smaller or bigger by one but never four or a multiple of four:
pi = 2 times 3/2 x 5/6 x 7/6 x 11/10 x 13/14 x 17/18 x
19/18 x 23/22 x 29/30 x 31/30 x 37/38 x 41/42 x 47/46 x
53/54 x 59/58 x 61/62 x 67/66 x 71/70 x 73/74 x ...
How did Euler find this product? via the zeta function?