  We prove the Central Limit Theorem (CLT), the first order Edgeworth Expansion
and a Mixing Local Central Limit Theorem (MLCLT) for Birkhoff sums of a class
of unbounded heavily oscillating observables over a family of full-branch
piecewise $C^2$ expanding maps of the interval. As a corollary, we obtain the
corresponding results for Boolean-type transformations on $\mathbb{R}$. The
class of observables in the CLT and the MLCLT on $\mathbb{R}$ include the real
part, the imaginary part and the absolute value of the Riemann zeta function.
Thus obtained CLT and MLCLT for the Riemann zeta function are in the spirit of
the results of Lifschitz & Weber (2009) and Steuding (2012) who have proven the
Strong Law of Large Numbers for "Sampling the Lindel\"of hypothesis".
