In the context of mod-Gaussian convergence, as defined previously in our work with J. 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 c...

Some positivity conjectures are made generalizing known results for Gaussian posets. For a sequence of non-negative integers ~ a = (a 1 ; a 2 ; ; a n), and a non-negative integer m, MacMahon 1, p. 137] considered the rational function of q,

V r /2 t,p/2 is a scaled version of Gaussian measure with r p if 1 < p < 2 and r p ′ if 2 < p < ∞ with 1/p 1/p′ 1. Conversely if F is a holomorphic function which is in a “slightly” smaller space, namely L C, V r/2 t,p/2 , then it is shown that there is a function f ∈ L R, ρ such that e−tHf F. However, a single necessary and sufficient condition is obtained for the image of L2 R, ρp/2 under e−t...