  In the present work we study the two-point correlation function $R(\epsilon)$
of the quantum mechanical spectrum of a classically chaotic system. Recently
this quantity has been computed for chaotic and for disordered systems using
periodic orbit theory and field theory. In this work we present an independent
derivation, which is based on periodic orbit theory. The main ingredient in our
approach is the use of the spectral zeta function and its autocorrelation
function $C(\epsilon)$. The relation between $R(\epsilon)$ and $C(\epsilon)$ is
constructed by making use of a probabilistic reasoning similar to that which
has been used for the derivation of Hardy -- Littlewood conjecture. We then
convert the symmetry properties of the function $C(\epsilon)$ into relations
between the so-called diagonal and the off-diagonal parts of $R(\epsilon)$. Our
results are valid for generic systems with broken time reversal symmetry, and
with non-commensurable periods of the periodic orbits.
