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2. Distribution of the Zeros of $\Lambda'_X(z)$.

    1. The Monodromy of the Derivative of Random Polynomials on the Unit Circle.

          Let \[P_N(z,w)=(z-w)\Lambda_X(z), \quad\quad |w|=1.\] Consider the equation $\frac{\partial}{\partial z}P_N(z,w)=0$. For each $w$, the solutions to this equation lie in the unit circle. We call these zeros the critical points.

      As we rotate $w$ around the unit circle, the critical points go on a trajectory before returning back to their original position. However, often in their trajectories, they permute with their neighbours.

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      Problem 2.1.

      [Chris Hughes and David Farmer] What can we understand about the cycle structure? In particular,
      • What happens to the permutation when the zeros of $\Lambda_X$ are the roots of unity?
      • If we have a zero close enough to the centre, is the permutation the identity?
        1. Remark. [org.aimpl.user:george.snape@bristol.ac.uk] As in the Double Hump problem, this question is a more general question for random polynomials with zeros on the unit circle. Does the cycle structure change if we draw the characteristic polynomials from the Poisson distribution, or the C$\beta$E distribution?

            • The Alternative Hypothesis and Zeros of the Derivative

                  An alternative to the pair correlation conjecture is the Alternative Hypothesis. Roughly speaking, it states that the normalised spacings between zeros of $\zeta$ are close to half-integers. In a 2019 blog post, Terence Tao introduced the ‘Alternative Circular Unitary Ensemble’, or ACUE. This is a probability ensemble of unitary matrices where the eigenangles are separated by a multiple of half the mean spacing. This is a probability distribution which closely resembles the CUE, whilst being consistent with the alternative hypothesis.

              Add remark

              Problem 2.2.

              [David Farmer] Can we understand the radial distribution of zeros of the derivative for the ACUE?

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