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.