The harmonic leading-digit law is one-sided at every N

Let P_d(N) be the probability that an integer drawn from {1, …, N} with weights proportional to 1/n has leading digit d in base b. It is classical that P_d(N) → p_d = log_b(1 + 1/d), Benford's law. We show that the deviation has a definite sign for every N: the likelihood ratio P_d(N)/p_d is strictly decreasing in d, the mantissa is strictly stochastically smaller than under Benford's law, and the Shannon entropy of the first k significant digits is strictly smaller than its Benford value, for every k. The same holds for the weights n^(−s), s > 1, also on all of ℕ. The only analytic input is the monotonicity of xψ′(x). We determine the liminf and limsup of the deviations normalized by ln N; for b = 10 the normalized entropy deficit oscillates between 0.2340… and 0.8876…. Finally, the one-sided law fails for the weights (ln n)^σ/n with σ > 0. In the profinite integers, with gcd(n, q) in place of the first digits and the totient entropy in place of the Benford entropy, the harmonic law is one-sided at every N too, and the normalized deficit converges.

The PDF contains the full listing of the verification script check_noteA.py (Appendix A) and the complete output of its run (Appendix B, 14/14 PASS); the archive contains the LaTeX source, the bibliography, the script and its output. The use of generative AI tools is described in the paper.

Author: Vladimir Khomyakov (Independent Researcher), ORCID 0009-0006-3074-9145. Version 1.0.
Concept-DOI (all versions): 10.5281/zenodo.23660785