I am curious as to whether the Erdos-Kac theorem can imply a lower bound on the density of primes, or perhaps imply the (weaker) infinitude of prime numbers all in all.
It seems as if the Erdos-Kac theorem should imply that the probability of having only $1$ prime factor should be roughly $1/\log n$, however, extrapolating this information from the assumption that the distribution is roughly Gaussian, and then calculating the probability of that event over the Gaussian random variable could lead one to think that the probability of having 0, or a negative number of prime divisors is also roughly $1/\log n$, which doesn't even make sense. Most likely, this implies that the convergence to Gaussian is slower in the distribution's tail, and that in particular there should be some error term roughly of that size.
Can anyone give me some direction as to how to think about the relation between the two theorems? And what kind of actual information can be extrapolated from the normal limit distribution, implied by the Erdos-Kac theorem, regarding the density of primes?
Ciao