On the derivatives of Hardy's function Z(t)

Let Z(k)(t) be the k-th derivative of Hardy’s Z-function. The numerics seem to suggest that if k and ℓ have the same parity, then the zeros of Z(k)(t) and Z(ℓ)(t) come in pairs which are very close to each other. That is to say that Z(k)(t)Z(ℓ)(t) has constant sign for the majority, if not almost all, of values t. In this paper we show that this is true a positive proportion of times. We also study the sign of the product of four derivatives of Hardy’s function, Z(k)(t)Z(ℓ)(t)Z(m)(t)Z(n)(t).