In this paper we use the Riemann zeta distribution to give a new proof of the Erdös-Kac Central Limit Theorem. That is, if
ζ
(
s
)
=
∑
n
≥
1
1
n
s
\zeta (s)=\sum _{n\ge 1} \frac {1}{n^s}
,
s
>
1
,
s>1,
then we consider the random variable
X
s
X_s
with
P
(
X
s
=
n
)
=
1
ζ
(
s
)
n
s
,
P(X_s=n)=\frac {1}{\zeta (s)n^s},
n
≥
1.
n\ge 1.
In an earlier paper, the first author and Adrien Peltzer derived the analog of the Erdös-Kac Central Limit Theorem (CLT) for the number of distinct prime factors,
ω
(
X
s
)
,
\omega (X_s),
of
X
s
,
X_s,
as
s
↘
1.
s\searrow 1.
In this paper we show, by means of a Tauberian Theorem, how to obtain the Central Limit Theorem of Erdös-Kac for the uniform distribution from the result for the random variable
X
s
X_s
. We also apply the technique to the number of distinct prime divisors of
X
s
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.