Let, as usual, \(Z(t)=e^{i\theta (t)}\zeta (1/2+it)\), where \(\theta (t)=\arg \pi^{-(1/4+it/2)} \Gamma (1/4+it/2).\) It is well known that on the Riemann hypothesis the zeros of \(Z(t)\) and those of \(Z'(t)\) are interlacing. In this paper a similar result is proved for \(Z'(t)\): if the Riemann hypothesis is true then there is a \(t_ 0>0\) such that for \(t>t_ 0\) the function \(Z''(t)\) has exactly one zero between consecutive zeros of \(Z'(t)\). In order to prove this result, the author introduces an analytic function whose zeros on the critical line are connected with those of \(Z'(t)\).