  For $X(n)$ a Steinhaus random multiplicative function, we study the maximal
size of the random Dirichlet polynomial $$ D_N(t) = \frac1{\sqrt{N}} \sum_{n
\leq N} X(n) n^{it}, $$ with $t$ in various ranges. In particular, for fixed
$C>0$ and any small $\varepsilon>0$ we show that, with high probability, $$
  \exp( (\log N)^{1/2-\varepsilon} ) \ll \sup_{|t| \leq N^C} |D_N(t)| \ll \exp(
(\log N)^{1/2+\varepsilon}). $$
