In the [paper](https://arxiv.org/abs/1902.07321) by Griffin, Ono, Rolen and Zagier which appeared on the arXiv today, (Update: published now in [PNAS](https://www.pnas.org/content/early/2019/05/20/1902572116)) the abstract includes



> 
> In the case of the Riemann zeta function, this proves the GUE random
>  matrix model prediction in derivative aspect. 
> 
> 
> 


In more detail, towards the bottom of the second page they say



> 
> Theorem 3 in the case of the Riemann zeta function is the *derivative
>  aspect Gaussian Unitary Ensemble* (GUE) random matrix model
>  prediction for the zeros of Jensen polynomials. To make this precise,
>  recall that Dyson, Montgomery, and Odlyzko ... conjecture that the
>  non-trivial zeros of the Riemann zeta function are distributed like
>  the eigenvalues of random Hermitian matrices. These eigenvalues
>  satisfy Wigner's Semicircular Law, as do the roots of the Hermite
>  polynomials $H\_d(X)$, when suitably normalized, as
>  $d\rightarrow+\infty$ ... The roots of $J\_{\gamma}^{d,0}(X)$, as
>  $d\rightarrow+\infty$, approximate the zeros of 
>  $\Lambda\left(\frac{1}{2}+z\right)$, ... and so GUE predicts that
>  these roots also obey the Semicircular Law. Since the derivatives of
>  $\Lambda\left(\frac{1}{2}+z\right)$ are also predicted to satisfy
>  GUE, it is natural to consider the limiting behavior of
>  $J\_{\gamma}^{d,n}(X)$ as $n\rightarrow+\infty$. The work here proves
>  that these derivative aspect limits are the Hermite polynomials
>  $H\_d(X)$, which, as mentioned above, satisfy GUE in degree aspect.
> 
> 
> 


I am hoping someone can further explain this. In particular, does this result shed any light on the horizontal distribution of the zeros of the derivative of the Riemann zeta function?




---


Edit: Speiser showed that the Riemann hypothesis is equivalent to $\zeta^\prime(s)\ne 0$ for $0<\sigma<1/2$. Since then quite a lot of work has gone into studying the horizontal distribution of the zeros of $\zeta^\prime(s)$. For example, [Duenez et. al.](https://arxiv.org/abs/1002.0372)[compared](https://iopscience.iop.org/article/10.1088/0951-7715/23/10/014) this distribution with the radial distribution of zeros
of the derivative of the characteristic polynomial of a random unitary matrix. (Caveat: I'm not up to date on all the relevant literature.)


This is a very significant question. If the GUE distribution holds for the Riemann zeros, then rarely but infinitely often there will be pair with less than than half the average (rescaled) gap. From this, by the work of Conrey and Iwaniec, one gets good lower bounds for the class number problem.


In [this](https://arxiv.org/abs/1002.1616) [paper](https://www.sciencedirect.com/science/article/pii/S0001870812001600) Farmer and Ki showed that if the derivative of the Riemann zeta function has sufficiently many zeros close to the
critical line, then the zeta function has many closely spaced zeros, which by the above, also solves the class number problem. 


The question of modeling the horizontal distribution of the zeros of $\zeta^\prime(s)$ with the radial distribution of zeros
of the derivative of the characteristic polynomial of a random unitary matrix, is intimately connected to the class number problem. Based on the answer of Griffin below, I don't think that's what the Griffin-Ono-Rolen-Zagier paper does, but it's worth asking about.



Jensen Polynomials for the Riemann Zeta Function
Just for clarification, what do you mean by the horizontal distribution? Are you asking about the distribution of the real parts of the zeros of $\zeta'(z):=\frac{\operatorname{d}}{\operatorname{d}s}\zeta(s)$? In that case, the answer is no. It is known that $\zeta'(s)$ does not satisfy the Riemann hypothesis, due to interaction with the trivial zeros, so the distribution of the real parts of these zeros is an interesting question, but not one on which I can comment. Our work only deals with the derivatives of the symmetric version of the zeta function, $\Lambda(z)$, which are all expected to satisfy both the Riemann hypothesis and the GUE model. If your question was about the distribution of these zeros, then yes, our theorem suggests that the low-lying zeros follow the GUE model with increasing accuracy as the order of the derivative increases.


