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

Abstract

Let $Z(t):=ζ\left(\frac{1}{2}+it\right)χ^{-\frac{1}{2}}\left(\frac{1}{2}+it\right)$ be Hardy's function, where the Riemann zeta function $ζ(s)$ has the functional equation $ζ(s)=χ(s)ζ(1-s)$. 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}-ε\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}-ε\right)\sqrt{\frac{\log T\log\log\log T}{\log\log T}}\right). \end{align*}