In the paper by Griffin, Ono, Rolen and Zagier which appeared on the arXiv today, 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 $0<\sigma<1$. 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.compared 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 paper 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.

In the paper by Griffin, Ono, Rolen and Zagier which appeared on the arXiv today, (Update: published now in PNAS) 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.compared 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 paper 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.

In the paper by Griffin, Ono, Rolen and Zagier which appeared on the arXiv today, 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?

In the paper by Griffin, Ono, Rolen and Zagier which appeared on the arXiv today, 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 $0<\sigma<1$. 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.compared 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 paper 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.