In this paper we use the Riemann zeta distribution to give a new proof of the Erdo<spacing diaeresis>s-Kac Central Limit Theorem. That is, if zeta(s) = Sigma(n >= 1) (1)(s)(n) , s > 1, then we consider the random variable X-s with P(X-s = n) = (1) (zeta) (s)n(s), n >= 1. In an earlier paper, the first author and Adrien Peltzer derived the analog of the Erdos-Kac Central Limit Theorem (CLT) for the number of distinct prime factors, omega(X-s), of X-s, as s SE arrow 1. In this paper we show, by means of a Tauberian Theorem, how to obtain the Central Limit Theorem of Erdos-Kac for the uniform distribution from the result for the random variable X-s. We also apply the technique to the number of distinct prime divisors of X-s that lie in an arithmetic sequence and a local CLT of the type proved by Dixit and Murty [Hardy-Ramanujan J. 43 (2020), 17-23] as well a version of the CLT for irreducible divisors of a monic polynomial over a finite field.