Jensen Polynomials for the Riemann Zeta Function

The GUE random matrix model predicts that the zeroes should satisfy the local statistics of random matrices. It doesn't predict that the zeroes should satisfy the global statistics of random matrices, because it's not clear what that would even mean unless the zeroes are all contained in some bounded interval. (In fact they get more frequent the further away from the origin).

The Hermite polynomials satisfy the global statistics of random matrices (the semicircular law) but not the local statistics (their zero spacing is very even). So it is not clear how a precise relationship between the GUE conjecture and this new theorem could be formulated.

For a toy model of what's going on here, we could take the function field model, where zeta functions look like $$ \frac{ \sum_{n=0}^{2g} a_n q^{-ns} }{ (1-q^{-s}) (1-q^{1-s} ) }.$$ The first step here is to multiply by a factor to make the functional equation as simple as possible, which for us is $q^{gs}$, and the second step is to multiply by a factor that kills the poles, which for us is $ (1-q^{-s}) (1-q^{1-s} )$. So the object being differentiated is $$\sum_{n=0}^{2g} a_n q^{(g-n) s}.$$ The $k$th derivative (with respect to $s$) is $$ (\log q)^k \sum_{n=0}^{2g} a_n (g-n)^k q^{g-ns}.$$ Renormalizing, we get $$ \sum_{n=0}^{2g} a_n \left(1 - \frac{n}{g} \right)^k q^{g-ns}.$$ As $k$ goes to $\infty$ and remains even, this converges to $$ a_0 q^{gs} + a_{2g} q^{-2g s},$$ which has all roots on the half-line, perfectly evenly spaced, since $$a_{2g} = q^g a_0$$ by the functional equation.

If we view these roots as lying on a circle (i.e. we take $q^{-s}$ for $s$ a root), we can say that they perfectly satisfy the global GUE statistics, being evenly distributed on the circle, but do not satisfy local GUE statistics, being perfectly evenly spaced. This is true regardless of whether our original zeta function behaved like a characteristic polynomial of a random matrix, or even whether it satisfied the Riemann hypothesis.

It is possible that a similar phenomenon is occurring for the Jensen polynomials of high derivatives.

Unfortunately, I cannot comment. https://www.youtube.com/watch?v=HAx_pKUUqug 56:28

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.