I performed some computations in wolfram alpha looking at the behavior of the values of $|\zeta(e^{ni})|$ trying to predict a lower bound. I have got the following result:

For $n > 19 :|\zeta(e^{ni})|\leq \log(n)$

**My question here is:**
Is the above result true and if it is, how can I show it?

**Note** :$i$ is the unit imaginary part of complex number and $n$ is a positive integer

Thank you for any help