  The original criteria of Riesz and of Hardy-Littlewood concerning the truth
of the Riemann Hypothesis (RH) are revisited and further investigated in light
of the recent formulations and results of Maslanka and of Baez-Duarte
concerning a representation of the Riemann Zeta function. Then we introduce a
general set of similar functions with the emergence of Poisson-like
distributions and we present some numerical experiments which indicate that the
RH may barely be true.
