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

TL;DR

This paper provides an unconditional asymptotic formula for the horizontal distribution of zeros of the derivative of the Riemann zeta-function, aligning with predictions from random matrix theory.

Contribution

It establishes a new asymptotic formula for the zeros of zeta'(s) in a specific range, confirming theoretical predictions.

Findings

Zeros of zeta'(s) are asymptotically distributed as predicted by random matrix theory.

The number of zeros with certain real parts and imaginary parts up to T follows a specific asymptotic formula.

The distribution of zeros matches the behavior of derivatives of characteristic polynomials of random unitary matrices.

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.