TITLE: Order of $\zeta(1+it)$
QUESTION [10 upvotes]: 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.]

REPLY [8 votes]: This a classical problem which has been worked on by Littlewood, Titchmarsh, Levinson, and more recently Granville & Soundararajan.
Littlewood showed that the Riemann Hypothesis (RH) implies
$$\limsup_{t\to \infty} \frac{|\zeta(1+it)|}{\log\log t} \leq 2e^\gamma$$
where $\gamma$ is Euler's constant. 
Titchmarsh and later Levinson showed (unconditionally) that 
$$\limsup_{t\to \infty} \frac{|\zeta(1+it)|}{\log\log t} \geq e^\gamma.$$
Hence, on RH, a factor of 2 remains unresolved. Littlewood conjectured that
$$\limsup_{t\to \infty} \frac{|\zeta(1+it)|}{\log\log t} = e^\gamma$$
but states "there is perhaps no good reason for believing ... this hypothesis."
A similar situation (a factor of 2 remaining unresolved) exits for large values of the function
$$1/|\zeta(1+it)|$$
when $|t|\geq 1$, say.
I got this information from the introduction to Granville and Soundararajan's paper: http://arxiv.org/abs/math/0501232. It can also be found in the notes of Chapter 13 in Montgomery & Vaughan's book "Multiplicative Number Theory." I believe it is also discussed in Heath-Brown's notes in the end of one of the chapters of Titchmarsh's book on the zeta-function.