Let $\zeta(s)$ denote the Riemann zeta-function. This thesis is concerned with the zeros of $\zeta’(s)$, the derivative of $\zeta(s)$. There are two main results established here. The first one is about the zero-counting problem for $\zeta’(s)$: 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 $0<\Im s\le T$. This is an analogue of Littlewood’s result on the zero-counting problem for $\zeta(s)$.

Our second result is a local structure theorem for zeros of $\zeta(s)$ and $\zeta’(s)$. Roughly speaking, we show that on the Riemann Hypothesis, if there occurs a small gap between consecutive zeta zeros, then there is exactly one zero of $\zeta’(s)$ lying not only very close to the critical line but also between that pair of zeta zeros. This refines a result of Zhang. As an application, we prove a stronger form of half of a conjecture of Radziwi{\l}{\l} concerning the global statistics of these zeros.

Some related results are also shown. For example, we prove a weak form of a conjecture of Soundararajan, and suggest a repulsion phenomena for zeros of $\zeta’(s)$.