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.
Categories: cond-mat