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arXiv · 1308.5116

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

S. J. Lester
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Abstract

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

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S. J. Lester. 2013-08-23. On the distribution of the zeros of the derivative of the Riemann zeta-function. https://doi.org/10.1017/s0305004114000413

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