I'm looking at an old paper of Montgomery, "Extreme values of the Riemann zeta function" and am having what I assume to be very rudimentary difficulties. This is the first time I've come across big Omega notation as opposed to big O notation. First of all, we have, for fixed $ \sigma \in \left[\frac{1}{2}, 1 \right) $ (and I presume for any small $ \epsilon > 0 $, probably picked so that $ 1 - \sigma - \epsilon > 0 $; this isn't really made clear)
$ \log \vert \zeta \left(\sigma + it \right) \vert = \Omega_+ ( \left(\log t \right)^{1 - \sigma - \epsilon}) $ as $ t \rightarrow \infty $.
It's my understanding that this means that $ \displaystyle \limsup_{t \rightarrow \infty} \frac{\log \vert \zeta \left(\sigma + it \right) \vert}{\left(\log t \right)^{1 - \sigma - \epsilon}} > 0 $.
Now the paper claims that the following is a sharper result:
$ \displaystyle \max_{1 \leq t \leq T} \log \vert \zeta \left(\sigma + it \right) \vert > c \frac{\left(\log T \right)^{1 - \sigma}}{\log \log T} $ for $ \sigma \in \left[\frac{1}{2}, 1 \right) $ and $ T \geq 10 $.
I understand that, for any small choice of $ \epsilon > 0 $, the right hand expression eventually dominates $ c \left(\log T \right)^{1 - \sigma - \epsilon} $ but aside from that I don't really understand how this result implies the first.
The lower bound $$\frac{\left(\log T \right)^{1 - \sigma}}{\log \log T}$$ is increasing for $T>T_0$ sufficiently large. For such $T$, if $t \in [1, T]$ with $$\log \vert \zeta \left(\sigma + it \right) \vert > c \frac{\left(\log T \right)^{1 - \sigma}}{\log \log T}$$ and if $c \frac{\left(\log T \right)^{1 - \sigma}}{\log \log T}$ is larger than $\max_{t \in [1,T_0]}\log \vert \zeta \left(\sigma + it \right) \vert$, we have $t > T_0$ and hence
$$ \log \vert \zeta \left(\sigma + it \right) \vert \geq c \frac{\left(\log t \right)^{1 - \sigma}}{\log \log t}$$
Finally, taking $T_0$ arbitrarily large gives arbitrarily large $t$. The weaker statement follows.