On some problems involving Hardy's function

TL;DR

This paper investigates various properties of Hardy's function, including its moments, value distribution, and primitive, providing insights into its behavior related to the Riemann zeta function.

Contribution

It explores new aspects of Hardy's function, such as odd moments and value distribution, extending understanding of its relation to the Riemann zeta function.

Findings

Analysis of odd moments of Z(t)

Distribution patterns of positive and negative values

Properties of the primitive of Z(t)

Abstract

Some problems involving the classical Hardy function $$ Z(t) := \zeta(1/2+it)\bigl(\chi(1/2+it)\bigr)^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s) $$ are discussed. In particular we discuss the odd moments of $Z(t)$, the distribution of its positive and negative values and the primitive of $Z(t)$. Some analogous problems for the mean square of $|\zeta(1/2+it)|$ are also discussed.