Hardy’s Z-function is defined as Z(t) =ζ(1/2+it)χ(1/2+it)^{-1/2},

where χ(s) is the factor from the functional equation for the zeta function,

ζ(s) =χ(s)ζ(1−s).

Essentially,Z(t) is a real valued version of the zeta function on the critical line, and Z(γ)= 0 if and only if ζ(1/2+iγ) = 0. Here we address the question of the proportion of t’s for which Z(t) is positive and negative. (This is joint work with A. Ivic).