We address the analysis of the following problem: given a real H\"older
potential $f$ defined on the Bernoulli space and $\mu_f$ its equilibrium state,
it is known that this shift-invariant probability can be weakly approximated by
probabilities in periodic orbits associated to certain zeta functions. Given a
H\"older function $f>0$ and a value $s$ such that $0<s<1$, we can associate a
shift-invariant probability $\nu_{s}$ such that for each continuous function
$k$ we have \[\int k d\nu_{s}=\frac{\sum_{n=1}^{\infty}\sum_{x\in
Fix_{n}}e^{sf^{n}(x)-nP(f)}\frac{k^{n}(x)}{n}}{\sum_{n=1}^{\infty}\sum_{x\in
Fix_{n}}e^{sf^{n}(x)-nP(f)}},\] where $P(f)$ is the pressure of $f$, $Fix_n$ is
the set of solutions of $\sigma^n(x)=x$, for any $n\in \mathbb{N}$, and
$f^{n}(x) = f(x) + f(\sigma(x)) + f(\sigma^2(x))+... + f(\sigma^{n-1} (x)).$ We
call $\nu_{s}$ a zeta probability for $f$ and $s$. It is known that $\nu_s \to
\mu_{f}$, when $s \to 1$. We consider for each value $c$ the potential $c f$
and the corresponding equilibrium state $\mu_{c f}$. What happens with
$\nu_{s}$ when $c$ goes to infinity and $s$ goes to one? This question is
related to the problem of how to approximate the maximizing probability for $f$
by probabilities on periodic orbits. We study this question and also present
here the deviation function $I$ and Large Deviation Principle for this limit
$c\to \infty, s\to 1$. We will make an assumption: $\lim_{c\to \infty, s\to 1}
c(1-s)= L>0$. We do not assume here the maximizing probability for $f$ is
unique.