  We demonstrate that beyond the universal regime correlators of quantum
spectral determinants $\Delta(\epsilon)=\det (\epsilon-\hat{H})$ of chaotic
systems, defined through an averaging over a wide energy interval, are
determined by the underlying classical dynamics through the spectral
determinant $1/Z(z)=\det (z- {\cal L})$, where $e^{-{\cal L}t}$ is the
Perron-Frobenius operator. Application of these results to the Riemann zeta
function, allows us to conjecture new relations satisfied by this function.
