On the distribution of positive and negative values of Hardy's Z-function

We investigate the distribution of positive and negative values of Hardy's functionZ(t):=ζ(12+it)χ(12+it)−1/2,ζ(s)=χ(s)ζ(1−s). In particular we prove thatμ(I+(T,T))≫Tandμ(I−(T,T))≫T, where μ(⋅) denotes Lebesgue measure andI+(T,H)={T0},I−(T,H)={T