除数加权Golomb序列：Zeta渐近与算术波动
A Divisor-Weighted Golomb Sequence: Zeta Asymptotics and Arithmetic Fluctuations
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中文总结 AI 辅助
本研究提出除数加权Golomb序列，证明其渐近指数仍为黄金比例，但常数改变，并揭示归一化游程长度具有奇异连续的算术波动分布。
中文摘要 AI 辅助
我们研究了Golomb自描述序列的一个除数加权版本：非递减序列$A$中，每个正整数$m$出现$\nsum_{d\mid m}A(d)$次。该规则具有唯一解，且$$ A(n)\sim Cn^{\varphi-1},\quad C=\left(\frac{\varphi}{\zeta(\varphi)}\right)^{2-\varphi},\quad \varphi=\frac{1+\sqrt{5}}{2}. $$ Golomb序列的黄金比例指数得以保留，而除数加权改变了首项常数。我们通过上下幂包络的收缩证明了该渐近性，无需假设正则变化。该论证适用于每个非负整数Dirichlet卷积核$w$，满足$w(1)=1$且$\sum_q w(q)q^{-\varphi}<\infty$。对于除数核，定量版本给出相对误差$O_\gamma((\log n)^{-\gamma})$，对每个$0<\gamma<1$成立。尽管全局增长是光滑的，归一化游程长度仍保留算术波动。与除数求和$\sum_{d\mid m}d^{1-\varphi}$的均匀比较将其转移到一个显式随机Euler积。我们证明其分布是奇异连续的，具有全支撑$[1,\infty)$，确定所有实数Mellin矩，获得波动的尖锐最大阶，并表明采样序列位置而非块标签会产生大小偏倚分布。
英文摘要
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.
