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 $\sum_{d\mid m}A(d)$ times. This rule has a unique solution, and $$ A(n)\sim Cn^{φ-1},\quad C=\left(\fracφ{ζ(φ)}\right)^{2-φ},\quad φ=\frac{1+\sqrt{5}}{2}. $$ 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 $\sum_q w(q)q^{-φ}<\infty$. 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 $\sum_{d\mid m}d^{1-φ}$ 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.