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 $$A(n)\sim C{n}^{\phi -1},C={\left(\frac{\phi }{\zeta (\phi )}\right)}^{2-\phi },\phi =\frac{1+\sqrt{5}}{2}.$$ 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 ${\sum }_{q}w(q)q^{-\phi }<\infty$. 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 ${\sum }_{d\mid m}{d}^{1-\phi }$ 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.

arXiv comment26 pages, 3 tables. Sequence: OEIS A400220