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 $(\log T{)}^{-1/3+\epsilon }\le (2\sigma -1)\le (\log \log T{)}^{-2}$, we show that the number of zeros of ${\zeta }^{\prime }(s)$ with imaginary part between zero and $T$ and real part larger than $\sigma$ is asymptotic to $T/(2\pi (\sigma -1/2))$ as $T\to \infty$. This agrees with a prediction from random matrix theory due to Mezzadri. Hence, for $\sigma$ in this range the zeros of ${\zeta }^{\prime }(s)$ are horizontally distributed like the zeros of the derivative of characteristic polynomials of random unitary matrices are radially distributed.

18 pages

References in corpus (1)

• Roots of the derivative of the Riemann zeta function and of characteristic polynomials

Cited by in corpus (1)

• On the critical points of random matrix characteristic polynomials and of the Riemann $\xi$-function