Gauss-Benford Rules and Their Harmonic Peers

Abstract

We research the relationship between the probability functions of the twofold hyperbolic universe, consisting of the logarithmic (real) and harmonic (rational) worlds. Within the logarithmic realm, we study the connection between the Gauss- Kuzmin distribution and Newcomb-Benford law and prove that they are fundamentally equivalent; the former corresponds to the probability decrements of the latter, i.e., log(2,1+1/(k(k+2))) is the difference between the function log(2,1+1/n) evaluated at n=k and n=k+1, where k is the index of a coefficient of a real number’s regular continued fraction expansion and n is a positive numeral written in positional notation. Thus, the binary Newcomb-Benford probability of n=1 is the sum of all the Gauss-Kuzmin masses, of n=2, is the sum of all the Gauss-Kuzmin masses minus the Gauss-Kuzmin mass at k=1, and of n=m, is the sum of all the Gauss-Kuzmin masses minus the sum of the first m-1 ones. Besides, the extrapolation of the Gauss-Kuzmin measure outside the unit interval subsumes Newcomb-Benford’s cumulative distribution function. These findings lead to the Gauss-Benford law, which specifies that the occurrence possibility of a positive real number represented in positional notation is log(2,1+x) (i.e., the Gauss-Kuzmin measure) if x is within the unit interval and log(2,1+1/x) if x is outside. Geometrically, these possibilities indicate proximity to 1. The map between both domain partitions is a sheer inversion, the unique conformal transformation that fixes one and respects the minimum information principle. Moreover, we introduce the Gauss-Benford measure, log(1+1/x,1+y), as the probability of a random variable accumulated between x-1 and x, with density 1/((1+y)ln(1+1/x)), where 0≤x-1<y≤x. We also explore the analogous harmonic rules; the canonical (normalized) harmonic probability of a positive natural q is 1/q, the qth harmonic gap has probability mass 1/(q(q+1)), and the harmonic occurrence possibility of a positive rational number is t if t is in the unit interval and 1/t if t is outside. We build the bridge between the logarithmic and harmonic realms by integrating and normalizing the latter.