How are these number-theoretical constants actually distributed?

Question

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.

Answer 1

These are Euler products which are convergent (though only conditionally, usually), and such Euler products quite often have a limiting distribution when taken in "reasonable" families. Once something like this is proved, one can conclude that the values are dense in the support of the limiting measure. These limit measures are most of the time characterized by the fact that their moments (or their Fourier transform) is itself an Euler product, the factors of which correspond to the limiting distribution of the $p$-component of the original family of constants. One can then hope to compute the support of the corresponding measure.

The classical example of doing this is for the values of the Riemann zeta-function, which goes back to Bohr and Jessen. It is treated in Chapter XI of Titchmarsh's book (and in that case, one can even deal with divergent Euler products). Another one, probably more directly related, is the study of the distribution of $L(1,\chi)$ for Dirichlet character $\chi$'s. There's been a lot of woork here, in particular recently of Granville and Soundararajan. A good survey is www.math.uiuc.edu/~lamzouri/DistribL2.pdf

Answer 2

This paper by Kowalski also looks highly relevant: http://arxiv.org/abs/0805.4682?context=math