公约数图的Havel--Hakimi残差：完整渐近展开与素数计数结构

中文摘要 AI 辅助

设$G_n$是顶点集为$\{2,\boldsymbol{\text{...}},n\boldsymbol{\rbrace}$的图，当两个整数的最大公约数大于1时相邻。我们确定其Havel--Hakimi残差$\boldsymbol{\text{R}}(G_n)$的完整渐近展开，确认Fajtlowicz《墙上书写》中记录的Staton的主项常数预测。若$A=\boldsymbol{\text{sum}}_{k=2}^{\boldsymbol{\text{infty}}}(\boldsymbol{\text{log}}k)/(k^2(k-1))$，则前两项为$\boldsymbol{\text{R}}(G_n)=(\boldsymbol{\text{zeta}}(2)-1)n/\boldsymbol{\text{log}}n+(\boldsymbol{\text{zeta}}(2)-1-A)n/\boldsymbol{\text{log}}^2n +O(n/\boldsymbol{\text{log}}^3n)$。更精确地说，对于每个固定的$0<\beta<1/2$，$\boldsymbol{\text{R}}(G_n)$与Caro--Wei和的素顶点贡献之差为$O_\beta(n\boldsymbol{\text{exp}}\boldsymbol{\text{\rbrace}})$，该估计给出展开中的所有系数。上界源于保度实现，其中几乎所有相关素顶点被划分为团。我们还证明，无标号图通过其单纯真孪生类确定$\boldsymbol{\text{pi}}(n)$。所得度类计数加权和的稳定逆给出等价于黎曼假设的判据，包括仅涉及Caro--Wei和的判据。精确的局部缺陷恒等式进一步将推测的严格$+2$残差界简化为显式前缀估计。

英文摘要

Let $G_n$ be the graph on $\{2,\ldots,n\}$ in which two integers are adjacent when they have a common divisor greater than one. We determine the complete asymptotic expansion of its Havel--Hakimi residue $\R(G_n)$, confirming a leading-constant prediction of Staton recorded in Fajtlowicz's \emph{Written on the Wall}. If $A=\sum_{k=2}^{\infty}(\log k)/(k^2(k-1))$, then the first two terms are $\R(G_n)=(ζ(2)-1)n/\log n+(ζ(2)-1-A)n/\log^2n +O(n/\log^3n)$. More precisely, the difference between $\R(G_n)$ and the prime-vertex contribution to the Caro--Wei sum is $O_β(n\exp\{-(\log n)^β\})$ for every fixed $0<β<1/2$; this estimate yields every coefficient in the expansion. The upper bound follows from a degree-preserving realization in which almost all relevant prime vertices are partitioned into cliques. We also prove that the unlabeled graph determines $π(n)$ through its simplicial true-twin classes. Stable inverses for weighted sums of the resulting degree-class counts give criteria equivalent to the Riemann hypothesis, including one involving only the Caro--Wei sum. An exact local-defect identity additionally reduces the conjectured sharp $+2$ residue bound to explicit prefix estimates.

发表机构

• FirstPrinciples Inc.（FirstPrinciples公司）
• Rice University（莱斯大学）

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