$\lim_{t \to \infty}\zeta(\frac{1}{2} + it)$

Question

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.

Answer 1

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...?

Answer 2

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.