Preprints.org.
ISSN/ISBN: Not available at this time. DOI: 10.20944/preprints202503.1625.v1
Abstract: We assume that the probability mass function Pr(Z) = (2Z)−2 (Z ∈ Z+) is at Newcomb-Benford Law’s root and the origin of positional notation. Under its tail, we find that the harmonic (global) Q-NBL for bijective numeration is Pr(b, q) = qHb −1, where q is a quantum (1 ≤ q ≤ b), Hn is the nth harmonic number, and b is the bijective base. Under its tail, the logarithmic (local) R-NBL for bijective numeration is Pr(r, d) = logr+1(1 + 1/d), where d ≤ r ≪ b, being d a digit of a local complex system’s bijective radix r. We generalize both lows to calculate the probability mass of the leading quantum/digit of a chain/numeral of a given length and the probability mass of a quantum/digit at a given position, verifying that the global and local NBL are length- and position-invariant in addition to scale-invariant. In the framework of bijective numeration, we also prove that the sums of Kempner’s series conform to the global Newcomb-Benford Law and suggest a natural resolution for the precision of a universal positional notation system.
Bibtex:
@misc{,
author = {Julio Rives},
title = {A Curious Effect of Benford’s Law for Bijective Numeration},
year = {2025},
url = {https://www.preprints.org/manuscript/202503.1625/v1},
}
Reference Type: Preprint
Subject Area(s): Probability Theory, Statistics