  We describe a new approach to constrain the amplitude of the power spectrum
of matter perturbations in the Universe, parametrized by sigma_8 as a function
of the matter density Omega_0. We compare the galaxy cluster X-ray luminosity
function of the REFLEX survey with the theoretical mass function of Jenkins et
al. (2001), using the mass-luminosity relationship obtained from weak lensing
data for a sample of galaxy clusters identified in Sloan Digital Sky Survey
commissioning data and confirmed through cross-correlation with the ROSAT
all-sky survey. We find sigma_8 = 0.38 Omega_0^(-0.48+0.27 Omega_ 0), which is
significantly different from most previous results derived from comparable
calculations that used the X-ray temperature function. We discuss possible
sources of systematic error that may cause such a discrepancy, and in the
process uncover a possible inconsistency between the REFLEX luminosity function
and the relation between cluster X-ray luminosity and mass obtained by Reiprich
& Bohringer (2001).
