# Inequalities regarding extreme values of the zeta function

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.