Energetics-spectral correlations versus the BATSE gamma-ray bursts population

Recently proposed correlations between the energetics of Gamma-Ray Bursts (GRB) and their spectral properties, namely the peak energy of their prompt emission, are found to account for the observed fluence distribution of all `bright' BATSE GRB. Furthermore for an intrinsic GRB peak energy distribution extending toward lower energies with respect to that characterizing bright GRB, such correlations allow to reproduce the fluence distribution of the whole BATSE long GRB population. We discuss the constraints that such analysis imposes on the shape of the peak energy distribution, the opening angle distribution and the tightness of such correlations.