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 $Z(t)$.