Is there a bounded function that is always greater than $M (t) = \max_{s \in [0, t]} \left| \zeta \left( \frac{1}{2} + i s \right) \right|$?

Question

Is there a bounded function that is always greater than $M (t) = \max_{s \in [0, t]} \left| \zeta \left( \frac{1}{2} + i s \right) \right|$ ?

Answer 1

It is not possible.

If true, there is some $C>0$ such that $|\zeta(1/2+it)|^2<C$. Then we would have $$\int_0^T |\zeta(1/2+it)|^2\, \text{d}t < CT.$$ However, it is well known that $$\int_0^T |\zeta(1/2+it)|^2\, \text{d}t \sim T\log T$$ so the two inequalites cannot hold simultaneously.

In fact it is possible to bound $\max_{T^\beta \leq t \leq T} |\zeta(1/2+it)|$ for any $0<\beta\leq 1$ by an unbounded function from below. In recent work by Bondarenko and Seip it is proved that for sufficiently large $T$, $$\max_{T^\beta \leq t \leq T} |\zeta(1/2+it)| \geq \exp\left(c\sqrt{\frac{\log T \log \log \log T}{\log \log T}}\right)$$ where $c$ is a positive number less than $\sqrt{1-\beta}$. Such results are commonly referred to as $\Omega$-results in the litterature, and dates back all the way to Titchmarsh.