Joint moments of derivatives of CUE characteristic polynomials and the Riemann zeta function
A talk in the Seminar Zufallsmatrizen series by
Alexander Grover
| Abstract: | We use Random Matrix Theory for the Circular Unitary Ensemble (CUE) to study joint integer moments of higher order derivatives of characteristic polynomials, motivated by moments of the Riemann zeta function. For finite lists $\mu$ and $\nu$, we consider mixed moments of the form \begin{equation}\mathbb{E}\left[\prod_i \Lambda_N^{(\mu_i)}(z)\prod_j \overline{\Lambda_N^{(\nu_j)}(z)}\right]. \end{equation} We first discuss the large-$N$ behaviour when $z$ lies suitably far inside the unit disc, where the leading term is expressed as an explicit combinatorial sum over contingency tables. This recovers the celebrated result of Diaconis--Gamburd at $z=0$, and we show how the same generating function arises in the corresponding problem for the Riemann zeta function near the critical line, conditionally on the Lindelöf hypothesis in full generality. We then present combinatorial formulae at finite $N$ and in the double-scaling regime, where the moments are given by sums of determinants weighted by Kostka numbers. This reveals the combinatorial structure produced by the distribution of derivative orders and allows us to recover a recent result of Simm and Wei as a special case. Using animations in Manim, the talk will highlight the key ideas behind these results from the recent preprint arXiv:2604.03051. Within the CRC this talk is associated to the project(s): C6 |