What is known about the order of $\zeta(1+it)$? 


Iwaniec-Kowalski gives (pp. 226 citing a result of Vinogradov-Korobov) 


$|\zeta(1+it)| \lesssim (\log t)^{2/3},$


and oppositely Titchmarsh gives (pp. 188 citing Bohr-Landau)


$|\zeta(1+it)| \gtrsim \log \log t$


for infinitely many values of $t$. 


Is this the limit of our knowledge? Is it  conditionally known (or even expected) unconditionally known that


$|\zeta(1+it)| = e^{o(\log \log t)}$?


[As David points out below, on RH the result in Titchmarsh's book is optimal.]



Order of $\zeta(1+it)$
The theorem quoted from I-K is due to, and should therefore be attributed to, Vinogradov-Korobov. It is the best known unconditional result on the line $\mathrm{Re}(s)=1$, and is directly related to the width of the zero-free region by a result of Landau (see around 3.10 in Titchmarsh), so that any improvement of either will give an improvement of the other, and then of the error term in the Prime Number Theorem.


