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

This is primarily an overview article on some results and problems involving the classical Hardy function $Z(t) := ζ(1/2+it){\bigl(χ(1/2+it)\bigr)}^{-1/2}, \quad ζ(s) = χ(s)ζ(1-s).$ 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.

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