In this question, we needed to compute the Hardy-Littlewood constant for the Bateman-Horn conjecture. Is there a simple argument to show that the infinite product actually converges? Also, is there a reasonable (for your favorite value of "reasonable") estimate of the convergence speed?
Well, the good news is that one can find an argument in the original paper of Bateman and Horn, which also gives the convergence rate, and which does not use Chebotarev. The bad news is that I don't understand why this works if the Galois group of $f$ is such that the average number of fixed points is not equal to $1.$