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1. Joint Moments of the Derivative of a Characteristic Polynomial

We will use the following notation. For $X\in U(N)$ and $z\in \mathbb{C}$, we denote the characteristic polynomial of $X$ by \[\Lambda_X(z)=\det(I-z X^\dagger),\] where $X^\dagger$ is the conjugate transpose of $X$. Furthermore, for a suitable function $f:U(N)\to\mathbb{C}$, we write \[\mathbb{E}_{U(N)}\left(f\right)=\int_{U(N)}f(X) dX,\] where we integrate over the Haar measure on $U(N)$.

There is now a vast literature on computing joint moments of the characteristic polynomial, which are defined to be the following average: \[\mathbb{E}_{U(N)}\left(|\Lambda_X(z)|^{2k-2h}|\Lambda_X'(z)|^{2h}\right),\] where $k,h$ and $z$ are complex numbers. However, the literature contains a multitude of different expressions, depending on the choice of restrictions of the parameters. For instance, one might specify that $k,h \in \mathbb{N}$ and $|z|=1$.

    1. Comparing the different formulations of the joint moments that exist in the literature


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      Problem 1.1.

      [Chris Hughes] A natural avenue to explore is to compare the strengths and deficiencies of the alternative formulations. For instance, one might ask:
      • Which formulations can give exact expressions for the joint moments for finite $N$?
      • Which method is best for a desired outcome?

        • Joint Moments Inside the Unit Circle

              This is a similar question to question $1$, but exploring joint moments and specifying that $|z|<1$. In particular, we ask:

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          Problem 1.2.

          • Can one derive an exact formula for the joint moments when $|z|=1-\frac{c}{N}$ and $k=h\in\mathbb{N}$, where $N$ is finite?
          • Can this be extended to $k\not= h\in\mathbb{N}$?

            • Joint Moments for $N=3$.


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              Problem 1.3.

              [Brian Conrey] In the case $N=2$, an exact formula for the joint moments where $k=h\in\mathbb{R}_{\geq 0}$ was established in a 2025 paper by Alvarez, Conrey, Rubinstein and Snaith (ACRS). Can one do an analogous computation for $N=3$?

                • Establishing Asymptotics from an Exact Formula for General $z\in\mathbb{C}$.


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                  Problem 1.4.

                  [Nick Simm] When $k=h\in\mathbb{N}$, Theorem 3.1 in the ACRS paper provides an exact formula for the joint moments, valid for all $z\in \mathbb{C}$. Can asymptotics of the formula be established to understand its large $N$ behavior?
                      In the case $|z|=1$, asymptotics are well understood. However, it would be useful to compare the different regimes.

                    • Painleve Representation for Joint Moments


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                      Problem 1.5.

                      [Brian Conrey] Does \[\mathbb{E}_U\left[|\Lambda_X'(1)|^{2k}\right]\] have a Painlevé representation for $k\notin \mathbb{N}$, in particular ‘non-classical’ Painlevé?
                        1. Remark. [org.aimpl.user:george.snape@bristol.ac.uk] There has been a recent breakthrough on this by Bothner and Wei.

                              Cite this as: AimPL: Moments of the derivative of characteristic polynomials and L-functions, available at http://aimpl.org/lprime.