I'm very curious about this and would be really grateful for any help or comments in this direction. If we consider any of the following number-theoretical constants:

1)The various singular series arising from any given system $\Psi: \mathbb{Z}^{d}\rightarrow \mathbb{Z}^t$,$d,t \geq 1$ in the Green-Tao paper "Linear Equations in Primes" (http://arxiv.org/PS_cache/math/pdf/0606/0606088v2.pdf):

$$ \prod_{p}\beta_p. $$

2)The Hardy-Littlewood/Bateman-Horn constants arising in the Hardy-Littlewood Conjecture F/ general Bateman-Horn conjecture, say for $f\in \mathbb{Z}[x]$ irreducible and $n_{p,f}$ the number of solutions to $f(n) \equiv 0 \bmod p$ in $\mathbb{Z}/p\mathbb{Z}$, the constant $$ C(f) = \prod_{p}\left(\frac{p-n_{p,f}}{p-1}\right), $$ and also the case (also covered by Bateman-Horn) where there is more than one polynomial (thus including the Hardy-Littlewood $k$-tuple conjecture)

3)The analogous constants that would arise naturally from maybe even a combination of (1) and (2) (Is this possible? I think the last slide of http://www.dpmms.cam.ac.uk/~bjg23/papers/icm-handout.pdf already talks about this.), so that one asks for systems $\Psi:\mathbb{Z}^d \rightarrow \mathbb{Z}^t$, $d,t\geq 1$, $\Psi = (\psi_1,\dots,\psi_t)$, where this time the $\psi_i$ need not be linear(!) (But here I'm only asking about the constants, not the actual conjecture.)

The question is, is there an easy way to see whether these constants are dense on any part of the real line or not? If I give you a fixed positive real number, and say you can pick any system from (3) that you want, or even something more general than (3), how would you go about systematically picking the system so that its constant is $\epsilon$-close to the given real number? Thank you very much.