[This paper](http://arxiv.org/pdf/math/0612106v2.pdf) p.4 claims:


Corollary C. Assume RH. For all large $t$ we have


$$|\zeta(\frac12 +it)| \le \exp\left(\frac38 \frac{\log{t}}{\log{\log{t}}}\right) \qquad (1) $$


$t$ a Gram points often appears counterexample to (1) according to mpmath, pari and maple. E.g. for $t=2381374874120.4, \, 352.4788 \not \le 24.2954 $


Other possible counterexamples $t$ are $$4992394.753, 42653554.76625,3293531640.5520, 29538618461.012969578$$ $$267653395649.1305498,2445999556058.1,2381374874120.4$$


The last one is $13$ digits.



> 
> How is "large" defined?
> 
> 
> 


**Added 2013-06-28** According to [Computations of the Riemann zeta function](http://sage.math.washington.edu/home/bober/www/data/zeta/) the inequality fails for $t \sim 3.925 \cdot 10^{31}$.


For $t=39246764589894309155251169284104.05199$, $\zeta(1/2 + it)=15837.8712 + 3604.9344i, |\zeta(1/2 + it)|=16242.95904$, while the bound from the inequality is $580.2737$ and $\log\log{t}=4.28$.



How is "large" defined in an equality for the modulus of Riemann zeta?
You could look at Chandee's paper (in Proc AMS) and on arxiv at <http://arxiv.org/pdf/0906.4177v1.pdf> where explicit bounds are worked out for zeta 
and L-function, and it is specified when they start to hold. For example 
see Corollary 1.4 there which holds for $t$ at least $\exp(\exp(10))$ which is about $10^{9500}$. 


