Hardy's function $Z(t)$ - results and problems

TL;DR

This paper reviews key results and open problems related to Hardy's function Z(t), including moments and value distribution, and introduces a new finding on its value distribution.

Contribution

It provides an overview of Hardy's function Z(t), discusses moments and value distribution, and presents a novel result on its value distribution.

Findings

Analysis of first and third moments of Z(t)

Distribution of positive and negative values of Z(t)

New result on the distribution of Z(t) values

Abstract

This is primarily an overview article on some results and problems involving the classical Hardy function $$ Z(t) := \zeta(1/2+it){\bigl(\chi(1/2+it)\bigr)}^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s). $$ In particular, we discuss the first and third moment of $Z(t)$ (with and without shifts) and the distribution of its positive and negative values. A new result involving the distribution of its values is presented.