For some more comments you can take a look at this question mathoverflow.net/questions/57981/… that discusses how hard it is just to determine if $r_2$ is zero or not.

Yes, for $n=p$ with $p$ prime your product is very close. For $n=2p$ it's larger by about $3/2$, for $n=3p$ by $8/7$, for $n=4p$ by $3/2$ again, for $n=5p$ by $24/23$, for $n=6p$ by $3/2\times8/7$. So based on the empirical evidence I'll conjecture $\frac{1}{2}\prod (1-1/p^2) \prod (1-2/q^2)$ over primes $p$ that divide $n$ and primes $q$ that don't. I don't have analysis to support it, but I look forward to your details.

I don't think MO is the right place for this question, maybe Stack Overflow stackoverflow.com is better.