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To treat the spectral statistics of quantum maps and flows that are fullychaotic classically, we use the rigorous Riemann-Siegel lookalike available forthe spectral determinant of unitary time evolution operators $F$. Concentratingon dynamics without time reversal invariance we get the exact two-pointcorrelator of the spectral density for finite dimension $N$ of the matrixrepresentative of $F$, as phenomenologically given by random matrix theory. Inthe limit $N\to\infty$ the correlator of the Gaussian unitary ensemble isrecovered. Previously conjectured cancellations of contributions ofpseudo-orbits with periods beyond half the Heisenberg time are shown to beimplied by the Riemann-Siegel lookalike.

Chaotic maps and flows: Exact Riemann-Siegel lookalike for spectral fluctuations