Havel--Hakimi Residues of Common-Divisor Graphs: Complete Asymptotics and Prime-Counting Structure

Abstract

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 $\R(G_n)=(\zeta(2)-1)n/\log n+(\zeta(2)-1-A)n/\log^2n +O(n/\log^3n)$. More precisely, the difference between $\R(G_n)$ and the prime-vertex contribution to the Caro--Wei sum is $O_\beta(n\exp\{-(\log n)^\beta\})$ for every fixed $0<\beta<1/2$; 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 $\pi(n)$ 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.