Large Positive and Negative Values of Hardy's $Z$-Function

Let $Z(t):=\zeta (\frac{1}{2}+it){\chi }^{-\frac{1}{2}}(\frac{1}{2}+it)$ 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 $ϵ>0$, \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*}