Mod-Gaussian convergence and the value distribution of $\zeta(1/2+it)$ and related quantities

In the context of mod-Gaussian convergence, as defined previously in our work with Jacod, we obtain asymptotic formulas and lower bounds for local probabilities for a sequence of random vectors which are approximately Gaussian in this sense, with increasing covariance matrix. 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.