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A Divisor-Weighted Golomb Sequence: Zeta Asymptotics and Arithmetic Fluctuations
Marco Mantovanelli
https://arxiv.org/abs/2610.03811 https://arxiv.org/pdf/2610.03811 https://arxiv.org/html/2610.03811
arXiv:2610.03811v1 Announce Type: new
Abstract: We study a divisor-weighted version of Golomb's self-description: the nondecreasing sequence $A$ in which each positive integer $m$ occurs $\sum_{d\mid m}A(d)$ times. This rule has a unique solution, and $$ A(n)\sim Cn^{\varphi-1},\quad C=\left(\frac{\varphi}{\zeta(\varphi)}\right)^{2-\varphi},\quad \varphi=\frac{1 \sqrt{5}}{2}. $$ 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 $\sum_q w(q)q^{-\varphi}<\infty$. For the divisor kernel, a quantitative version gives relative error $O_\gamma((\log n)^{-\gamma})$ for every $0<\gamma<1$.
Although the global growth is smooth, the normalized run lengths retain arithmetic fluctuations. A uniform comparison with the divisor sum $\sum_{d\mid m}d^{1-\varphi}$ transfers them to an explicit random Euler product. We prove that its law is singular continuous with full support $[1,\infty)$, 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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🤖 04/08 10:01

Bin zwar später losgekommen als geplant, da das Richten und Packen von zum Wetterbericht passenden Klamotten — hatte die komplette Regenmontur dabei — doch mehr Zeit in Anspruch nahm, aber ansonsten ging der Plan auf: Bin gestern mit 'm #BromptonElectricGLine zur #MiniDebConfWinterthur

 Iran-aligned Houthis seized control of Yemen’s port city of Mocha on Thursday
and advanced down the Red Sea coast to strategic islands, military sources said,
hours after Donald Trump said he expected the Iran war to end after midterm elections.
Yemeni government military sources said the Houthis had gained further leverage over the Bab el-Mandeb Strait,
the southern outlet of the Red Sea and one of the world’s most important shipping routes,
and had reached t…

Houthis Seize Mocha, Tightening Grip on Bab el-Mandeb and Red Sea Shipping
Iran-aligned Houthis seized control of Yemen's port city of Mocha on Thursday, military sources said, posing a further threat to global energy supplies hours after President Donald Trump said he expected the war to end after mid-term elections.

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