On large values of Hardy’s function $Z(t)$ and its derivatives

Let $Z(t) = \zeta(\frac{1}{2}+it)\chi^{-1/2}(\frac{1}{2}+it)$ denote as usual Hardy’s function, where $\zeta(s) = \chi(s)\zeta(1-s)$ is the functional equation for the Riemann zeta-function $\zeta(s)$. It is proved that, for $t\ge t_0 \gt 0$, \begin{gather*} \max\limits_{T\le t\le T+H, Z(t) \gt 0}Z(t) \gg ({\rm log}\, T)^{1/4}\qquad(T^{\theta+\varepsilon}\le H \le T),\\ \max\limits_{T\le t\le T+H, Z(t) \lt 0}-Z(t) \gg ({\rm log}\, T)^{1/4}\qquad(T^{\theta+\varepsilon}\le H \le T), \end{gather*} where $\theta = \frac{17}{110} = 0.15\overline{45}$. A similar result is shown for large values of $Z^{(k)}(t)$, where $k\ge1$ is a fixed integer. Several related topics are also discussed.