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

TL;DR

This paper establishes new lower bounds for the maximum positive and negative values of Hardy's Z-function within certain ranges, revealing extremely large oscillations related to the growth of the zeta function.

Contribution

It proves that Hardy's Z-function attains extraordinarily large positive and negative values in specified intervals, advancing understanding of its extreme oscillations.

Findings

Hardy's Z-function has large positive values in certain ranges.

Hardy's Z-function attains large negative values in certain ranges.

The bounds involve exponential functions of logarithmic terms.

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 }& \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*}