  In the context of mod-Gaussian convergence, as defined previously in our work
with J. Jacod, we obtain lower bounds for local probabilities for a sequence of
random vectors which are approximately Gaussian with increasing covariance.
This is motivated by the conjecture concerning the density of the set of values
of the Riemann zeta function on the critical line. We obtain evidence for this
fact, and derive unconditional results for random matrices in compact classical
groups, as well as for certain families of L-functions over finite fields.
