  Let $Z(t)$ be the classical Hardy function in the theory of the Riemann
zeta-function. The main result in this paper is that if the Riemann hypothesis
is true then for any positive integer $n$ there exists a $t_{n}>0$ such that
for $t>t_{n}$ the function $Z^{(n+1)}(t)$ has exactly one zero between
consecutive zeros of $Z^{(n)}(t)$.
