  Let
$Z(t):=\zeta\left(\frac{1}{2}+it\right)\chi^{-\frac{1}{2}}\left(\frac{1}{2}+it\right)$
be Hardy's function, where the Riemann zeta function $\zeta(s)$ has the
functional equation $\zeta(s)=\chi(s)\zeta(1-s)$. We prove that for any
$\epsilon>0$, \begin{align*} &\quad\max_{T^{3/4}\leq t\leq T} Z(t) \gg
\exp\left(\left(\frac{1}{2}-\epsilon\right)\sqrt{\frac{\log T\log\log\log
T}{\log\log T}}\right)\\ \text{ and }& \max_{T^{3/4}\leq t\leq T}- Z(t) \gg
\exp\left(\left(\frac{1}{2}-\epsilon\right)\sqrt{\frac{\log T\log\log\log
T}{\log\log T}}\right). \end{align*}
