$\lim_{t \to \infty}\zeta(\frac{1}{2} + it)$ I am learning about to zeta-function, I am a beginner.
I am trying to find:
$$\lim_{t \to \infty}\zeta(\frac{1}{2} + it)$$
From the book Theory of the Riemann zeta-function-clarendon by Titchmarsh in the theorem 8.12
I know:
If $\frac{1}{2} \leq \sigma < 1$ the $|\zeta(\sigma + it)|> e^{\log^{\alpha}y}$ with $\alpha < 1- \sigma $ and for some indefinitely large values of $t$
For my case if $0<\alpha<\frac{1}{2}$ then since $ |\zeta(1/2 + it)|> e^{\log^{\alpha}t}$ for some indefinitely large values of $t$ since $t \to \infty$ then I don't know if that's enough to conclude that $$\lim_{t \to \infty}\zeta(\frac{1}{2} + it) $$ does not converge.
Please any help is good.
 A: As @Gary comments, there is some ill-definedness and muddle in the statements of the question, and the supposedly-cited facts.
One very clear point is that it is known by now that there are infinitely-many zeros on the critical line (Levinson... Conrey... showed that at least 2/5 (?) or so are on-the line...), but/and away from zeros zeta grows. So there's no actual limit of $\zeta({1\over 2}+it)$ as $t\to \infty$.
Naturally, with or without RH, things are even more chaotic to the right of $\Re(s)={1\over 2}$. For example, Voronin's universality theorem overwhelmingly indicates that there are no elementary asymptotics on any vertical line to the right of $\Re(s)={1\over 2}$. Overkill, yes, but really decisive.
But/and, perhaps your true question can be more refined...?
A: By Titchmarsh's theorem,
$$
\mathop {\lim \sup }\limits_{t \to  + \infty } \,\left| {\zeta\! \left( {\tfrac{1}{2} + it} \right)} \right| =  + \infty .
$$
We also know that there is an infinite number of zeros along the critical line (with an accumulation point at infinity), so
$$
\mathop {\lim \inf }\limits_{t \to  + \infty } \,\left| {\zeta\! \left( {\tfrac{1}{2} + it} \right)} \right| = 0.
$$
Since the two are not equal, the limit $\lim_{t \to  + \infty } \left| {\zeta\! \left( {\tfrac{1}{2} + it} \right)} \right|$ does not exist, and hence the limit $\lim_{t \to  + \infty } \zeta\! \left( {\tfrac{1}{2} + it} \right)$ does not exist.
