  In recent studies of many-body localization in nonintegrable quantum systems,
the distribution of the ratio of two consecutive energy level spacings,
$r_n=(E_{n+1}-E_n)/(E_{n}-E_{n-1})$ or $\tilde{r}_n=\min(r_n,r_n^{-1})$, has
been used as a measure to quantify the chaoticity, alternative to the more
conventional distribution of the level spacings,
$s_n=\bar{\rho}(E_n)(E_{n+1}-E_n)$, as the former unnecessitates the unfolding
required for the latter. Based on our previous work on the Tracy-Widom approach
to the Janossy densities, we present analytic expressions for the joint
probability distribution of two consecutive eigenvalue spacings and the
distribution of their ratio for the Gaussian unitary ensemble (GUE) of random
Hermitian $N\times N$ matrices at $N\to\infty$, in terms of a system of
differential equations. As a showcase of efficacy of our results for
characterizing an approach to quantum chaoticity, we contrast them to arguably
the most ideal of all quantum-chaotic spectra: the zeroes of the Riemann
$\zeta$ function on the critical line at increasing heights.
