On some problems involving Hardy's function

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.