Moments of characteristic polynomials at random matrix Gram points

Abstract

We evaluate the characteristic polynomial of a random unitary matrix at the points on the unit circle where the polynomial takes real values. In analogy to the Riemann zeta function we call these the ``random matrix Gram points''. On the random matrix side we answer in the affirmative a question of Richard Hall, who expected that zeta on the Gram points should be larger than typical. We prove that for any fixed positive $k$, the $2k$th moment of the characteristic polynomial averaged over the random matrix Gram points is larger than the $2k$th moment of the characteristic polynomial averaged over the whole unit circle. For large $N$ the leading-order asymptotic term is the same size, but all lower-order terms are larger. In an appendix we show how to adapt the heuristic recipe of Conrey–Farmer–Keating–Rubinstein–Snaith to conjecture moments of zeta at Gram points, and show by example that the second and fourth moments calculated by following the recipe agree with previously proven results.