A Divisor-Weighted Golomb Sequence: Zeta Asymptotics and Arithmetic Fluctuations
Abstract
We study a divisor-weighted version of Golomb's self-description: the nondecreasing sequence A in which each positive integer m occurs ∑d| mA(d) times. This rule has a unique solution, and A(n)∼ Cnφ−1, C=(φ/ζ(φ))2−φ, φ=(1+√5)/2. The golden-ratio exponent of Golomb's sequence survives, while divisor weighting changes the leading constant. We prove the asymptotic without assuming regular variation, by a contraction of upper and lower power envelopes. The argument applies to every nonnegative integer Dirichlet-convolution kernel w with w(1)=1 and ∑q w(q)q−φ<∞. For the divisor kernel, a quantitative version gives relative error Oγ((log n)−γ) for every 0<γ<1. Although the global growth is smooth, the normalized run lengths retain arithmetic fluctuations. A uniform comparison with the divisor sum ∑d| md1−φ transfers them to an explicit random Euler product. We prove that its law is singular continuous with full support [1,∞), determine all real Mellin moments, obtain the sharp maximal order of the fluctuations, and show that sampling sequence positions instead of block labels produces the size-biased law.
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BibTeX
@misc{mantovanelli2026divisorweighted,
title = {{A Divisor-Weighted Golomb Sequence: Zeta Asymptotics and Arithmetic Fluctuations}},
author = {Marco Mantovanelli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03811},
eprint = {2610.03811},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03811},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Divisor-Weighted Golomb Sequence: Zeta Asymptotics and Arithmetic Fluctuations
AU - Marco Mantovanelli
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.03811
UR - https://arxiv.org/abs/2610.03811
AB - We study a divisor-weighted version of Golomb's self-description: the nondecreasing sequence A in which each positive integer m occurs ∑_d| mA(d) times. This rule has a unique solution, and A(n)∼ Cn^φ−1, C=(φ/ζ(φ))^2−φ, φ=(1+√5)/2. The golden-ratio exponent of Golomb's sequence survives, while divisor weighting changes the leading constant. We prove the asymptotic without assuming regular variation, by a contraction of upper and lower power envelopes. The argument applies to every nonnegative integer Dirichlet-convolution kernel w with w(1)=1 and ∑_q w(q)q^−φ<∞. For the divisor kernel, a quantitative version gives relative error O_γ((log n)^−γ) for every 0<γ<1. Although the global growth is smooth, the normalized run lengths retain arithmetic fluctuations. A uniform comparison with the divisor sum ∑_d| md^1−φ transfers them to an explicit random Euler product. We prove that its law is singular continuous with full support [1,∞), determine all real Mellin moments, obtain the sharp maximal order of the fluctuations, and show that sampling sequence positions instead of block labels produces the size-biased law.
ER -