  We improve Montgomery's $\Omega$-results for $|\zeta(\sigma+it)|$ in the
strip $1/2<\sigma<1$ and give in particular lower bounds for the maximum of
$|\zeta(\sigma+it)|$ on $\sqrt{T}\le t \le T$ that are uniform in $\sigma$. We
give similar lower bounds for the maximum of $|\sum_{n\le x} n^{-1/2-it}|$ on
intervals of length much larger than $x$. We rely on our recent work on lower
bounds for maxima of $|\zeta(1/2+it)|$ on long intervals, as well as work of
Soundararajan, G\'{a}l, and others. The paper aims at displaying and clarifying
the conceptually different combinatorial arguments that show up in various
parts of the proofs.
