  We propose that the widely observed and universal Gutenberg-Richter relation
is a mathematical consequence of the critical branching nature of earthquake
process in a brittle fracture environment. These arguments, though preliminary,
are confirmed by recent investigations of the seismic moment distribution in
global earthquake catalogs and by the results on the distribution in crystals
of dislocation avalanche sizes. We consider possible systematic and random
errors in determining earthquake size, especially its seismic moment. These
effects increase the estimate of the parameter beta of the power-law
distribution of earthquake sizes. In particular, we find that estimated
beta-values may be inflated by 1-3% because relative moment uncertainties
decrease with increasing earthquake size. Moreover, earthquake clustering
greatly influences the beta-parameter. If clusters (aftershock sequences) are
taken as the entity to be studied, then the exponent value for their size
distribution would decrease by 5-10%. The complexity of any earthquake source
also inflates the estimated beta-value by at least 3-7%. The centroid depth
distribution also should influence the beta-value, an approximate calculation
suggests that the exponent value may be increased by 2-6%. Taking all these
effects into account, we propose that the recently obtained beta-value of 0.63
could be reduced to about 0.52--0.56: near the universal constant value (1/2)
predicted by theoretical arguments. We also consider possible consequences of
the universal beta-value and its relevance for theoretical and practical
understanding of earthquake occurrence in various tectonic and Earth structure
environments. Using comparative crystal deformation results may help us
understand the generation of seismic tremors and slow earthquakes and
illuminate the transition from brittle fracture to plastic flow.
