I performed some computations in [wolfram alpha](https://www.wolframalpha.com/input/?i=%7Czeta%5Be%5E(174521i))%5D%7C)%5Cleq+log(174521)) 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 



Is $|\zeta(e^{ni})|\leq \log(n)$ true for $n > 19$ and how do i can show it if it is?
The minimal counterexample is $n=25$ (see the [numerical verification](https://www.wolframalpha.com/input/?i=%7Czeta%5Be%5E(25i))%5D%7C)%5Cleq+log(25))).


Another small counterexample is $n=44$, which as suggested by Gerry Myerson comes from the approximation $2\pi \approx 44/7$.


