In this paper, we study the random field \begin{equation*} X(h) \circeq
\sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1],
\end{equation*} where (Up,pprimes) is an i.i.d. sequence of
uniform random variables on the unit circle in C. Harper (2013)
showed that (X(h),h∈(0,1)) is a good model for the large values of
(log∣ζ(21+i(T+h))∣,h∈[0,1]) when T is large, if
we assume the Riemann hypothesis. The asymptotics of the maximum were found in
Arguin, Belius & Harper (2017) up to the second order, but the tightness of the
recentered maximum is still an open problem. As a first step, we provide large
deviation estimates and continuity estimates for the field's derivative
X′(h). The main result shows that, with probability arbitrarily close to 1,
\begin{equation*} \max_{h\in [0,1]} X(h) - \max_{h\in \mathcal{S}} X(h) = O(1),
\end{equation*} where S a discrete set containing O(logTloglogT) points.Comment: 7 pages, 0 figur