  The temperature distribution function (TDF) of X-ray clusters is derived for
different normalizations of the HDM power spectrum in an Omega_0=1 universe
using the statistics of peaks of a random Gaussian field. Concerning cluster
formation only, the neutrino picture appears to be marginally consistent with
observations for H_0=70 km/s/Mpc provided the normalization is
alpha=sigma_0/delta_s=0.42 with sigma_0 the rms variance of the fluctuations
linearly evolved to the present epoch and delta_s the collapse threshold
required in structure formation models. However, using the constraint that the
formation epoch of QSO's puts on the HDM model, it appears that this
normalization is too small to be consistent with quasars formation. Allowing
higher values for the Hubble constant alleviates this problem but leads to an
unacceptable short age of the universe. Moreover, the constraint put by the
recent measurement of the level of fluctuations in the cosmic background
radiation by COBE tends to reject models with high values of H_0. We conclude
that the neutrino model fails in any case to account for the whole set of
available observations in a consistent way.
