Number theory, borderline dimension and extensive entropy in distributions of ranked data

Abstract

The consideration of an existing stochastic approach for the reproduction of ranked data pointed at a formal equivalence between its key mathematical expression and that for trajectories at the tangent bifurcation. This fact led to a nonlinear dynamical approach for rank distributions that shows similarities with universality classes in critical phenomena. The renormalization group (RG) fixed-point map f*(x) for a tangent bifurcation of arbitrary nonlinearity z > 1 has proved to be a powerful tool into which the formalism can be couched. The source distribution P(N) of the stochastic approach can be linked to f*(x) while the size-rank N(k) and frequency-rank F(k′) distributions are obtained, respectively, from the map trajectories xt and the sums of its positions. We provide now an extension to Number Theory as we obtain from the trajectories xt of f*(x) the numbers, or asymptotic approximations of them, for the Factorial, Natural, Prime and Fibonacci sets. A measure of the advance of these numbers towards infinity is given by sums of positions that represent their reciprocals. We specify rank distribution universality classes, already associated with real data, to these number sets. We find that the convergence of the series of number reciprocals occurs first at nonlinearity z = 2, that which corresponds to the classical Zipf law, and link this transition edge to the action of the attractor when it first reduces the fractal dimension of trajectory positions to zero. Furthermore, the search of logarithmic corrections common to borderline dimensions provides a link to the Prime numbers set. Finally, we find corroborating evidence of these logarithmic corrections from the analysis of large data sets for ranked earthquake magnitudes. The formalism links all types of ranked distributions to a generalized extensive entropy.