  New aspects of spectral fluctuations of (quantum) chaotic and diffusive
systems are considered, namely autocorrelations of the spacing between
consecutive levels or spacing autocovariances. They can be viewed as a
discretized two point correlation function. Their behavior results from two
different contributions. One corresponds to (universal) random matrix
eigenvalue fluctuations, the other to diffusive or chaotic characteristics of
the corresponding classical motion. A closed formula expressing spacing
autocovariances in terms of classical dynamical zeta functions, including the
Perron-Frobenius operator, is derived. It leads to a simple interpretation in
terms of classical resonances. The theory is applied to zeros of the Riemann
zeta function. A striking correspondence between the associated classical
dynamical zeta functions and the Riemann zeta itself is found. This induces a
resurgence phenomenon where the lowest Riemann zeros appear replicated an
infinite number of times as resonances and sub-resonances in the spacing
autocovariances. The theoretical results are confirmed by existing ``data''.
The present work further extends the already well known semiclassical
interpretation of properties of Riemann zeros.
