The Number of Zeros of ${\zeta }^{\mathrm{\prime }}(s)$

Assuming the Riemann Hypothesis, we prove that $$N_1(T)=\frac{T}{2\pi }\log \frac{T}{4\pi e}+O\left(\frac{\log T}{\log \log T}\right),$$ where $N_1(T)$ is the number of zeros of ${\zeta }^{\mathrm{\prime }}(s)$ in the region $0<\mathrm{\Im }s\le T$.