Preprint on ResearchGate.
ISSN/ISBN: Not available at this time. DOI: 10.5281/zenodo.18686437
Abstract: Benford’s Law predicts that first significant digits in many datasets follow P (d) = log10(1+1/d). We develop a number-theoretic framework explaining when and why this law is violated. The Digit Periodicity Theorem shows that the first-digit sequence of {n−k}k≥1 has approximate period q whenever log10(n) is well approximated by a rational p/q, with error bounded by qε where ε = | log10(n) − p/q|. We define the logarithmic resonance Rlog(n), a measure of rational approximation quality for log10(n), and prove that Rlog(n) = 0 if and only if n is a power of 10. Via Weyl’s equidistribution theorem, we establish that {n−k} follows Benford’s Law asymptotically for all n that are not powers of 10, and that the convergence rate is controlled by Rlog(n). We introduce the Structure Index ψ, a chi-squared measure of Benford deviation, and derive significance thresholds from the χ2(8) distribution. For sequences governed by inverse power laws f (n) = 1/nk, we derive an explicit first-digit distribution Pk(d) and prove that the chi-squared deviation from Benford is monotonically decreasing in k. We establish that classical Benford testing must reject authentic power-law data above a computable sample size, prove the formula is asymptotically exact for finite samples, prove that each power-law exponent produces a unique first-digit fingerprint enabling consistent identification of the underlying exponent from data, derive a contamination detection theorem (ψmix = α2ψk for mixture fraction α), and apply Baker’s theorem to prove an effective lower bound Rlog(p) > 0 for all primes.
Bibtex:
@misc{,
author = {Francisco Bustillos},
title = {Digit Periodicity, Benford’s Law, and Logarithmic Resonance},
year = {2026},
doi = {10.5281/zenodo.18686437},
url = {https://www.researchgate.net/profile/Francisco-Bustillos-Morales/publication/400961288_DIGIT_PERIODICITY_BENFORD'S_LAW_AND_LOGARITHMIC_RESONANCE/links/69975ffa42f94d1212ac0c0b/DIGIT-PERIODICITY-BENFORDS-LAW-AND-LOGARITHMIC-RESONANCE.pdf},
}
Reference Type: Preprint
Subject Area(s): Number Theory, Statistics