The Erdos-Kac Central Limit Theorem says that if one selects an integer at random from 1 to N, then the number of distinct prime divisors of this number satisfies a Central Limit Theorem. We (the speaker in joint work with Tom Mountford) give new proof of this result using the Riemann zeta distribution and a Tauberian Theorem. The proof generalizes easily to other situations such as polynomials over a finite field or ideals in a number field.