  Assuming the Riemann Hypothesis, we prove that $$ N_1(T) = \frac{T}{2\pi}\log
\frac{T}{4\pi e} + O\bigg(\frac{\log T}{\log\log T}\bigg), $$ where $N_1(T)$ is
the number of zeros of $\zeta'(s)$ in the region $0<\Im s\le T$.
