Large positive and negative values of Hardy’s $Z$-function

Abstract:

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}\quad &\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 ). \end{align*}