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}-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*}

8 pages, some minor changes, To appear in Proceedings of the American Mathematical Society

References in corpus (1)

• Extreme values of the Riemann zeta function and its argument