Universality of Riemann zeta function , Which related to the approximation of every Holomorphic function $f(z)$ by Riemann zeta function in the strip , it states for lower density:
Corollary: Let $K_0$ be a compact set in the right half of the critical stripe $1/2< \Re z<1$. Let $f$ be a continuous function on $K_0$, which is holomorphic on an open set containing $K_0$ and does not have any zeros in $K_0$ . For every $\epsilon_0>0$, we have that the limit (lower density ) $$ \inf\lim\limits_{T \rightarrow \infty} \frac{1}{T} \lambda \Big (\{ t\in[0,T]: \max\limits_{z \in K_0} \left| {\zeta(z+it) -f(z) )}\right| < \epsilon_0\Big\}) $$ is positive for $\lambda$ being the Lebesgue measure.
Now my question here : Can we approximate Riemann zeta function in the helf strip by the same strategy by means using that Corollary (lower density) since it is holomorphic ?
Note:The part of this question is montioned as an answer here