Derivation of the Gutenberg-Richter Empirical Formula from the Solution   of the Generalized Logistic Equation
  We have written a new equation to study the statistics of earthquake distributions. We call this equation "the generalized logistic equation". The Gutenberg-Richter frequency-magnitude formula was derived from the solution of the generalized logistic equation as an asymptotic case in approximation of large magnitudes. To illustrate how the found solution of the generalized logistic equation works, it was used to approximate the observed cumulative distribution of earthquakes in four different geological provinces: the Central Atlantic (40N-25N, 5W-35W), Canary Islands, Magellan Mountains (20N-9S, 148E-170E), and the Sea of Japan. This approximation showed the excellent fit between the theoretical curves and observed data for earthquake magnitudes 1<m<9. 
Lev A. Maslov, Vladimir M. Anokhin
physics.geo-ph