Sums and products indexed by primes are very well studied. Look up Legendre's method and its related improvements for calculating pi(x), see also the Riemann Zeta function.
You were at least attempting to apply one technique from another area in number theory. That is how a lot of research gets done. As is the case with most first ideas it doesn't lead anywhere. Feynman is supposed to have given a talk saying 'hey, thanks for the prize for this really great idea I had, here are the 20 that came before it that were all stupid' or words to that effect.