On the distribution of the zeros of the derivative of the Riemann zeta-function

Abstract

We establish an unconditional asymptotic formula describing the horizontal distribution of the zeros of the derivative of the Riemann zeta-function. For $\Re(s)=\sigma$ satisfying $(\log T)^{-1/3+\epsilon} \leq (2\sigma-1) \leq (\log \log T)^{-2}$, we show that the number of zeros of $\zeta'(s)$ with imaginary part between zero and $T$ and real part larger than $\sigma$ is asymptotic to $T/(2\pi(\sigma-1/2))$ as $T \rightarrow \infty$. This agrees with a prediction from random matrix theory due to Mezzadri. Hence, for $\sigma$ in this range the zeros of $\zeta'(s)$ are horizontally distributed like the zeros of the derivative of characteristic polynomials of random unitary matrices are radially distributed.