Hardy’s function Z(t): Results and problems

Abstract

This is primarily an overview article on some results and problems involving the classical Hardy function Z(t):= ζ(1/2 + it)(χ(1/2 + it))^−1/2, ζ(s) = χ(s)ζ(1 − s). In particular, we discuss the first and third moments 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.