An Average-Order Theorem for a Shifted Pairwise-Coprime Extremal Problem

Abstract

For $n\ge 2$, let $\mathcal{M}(n)$ be the supremum of $\sum_{a\in A}1/(n-a)$ over pairwise coprime sets $A\subset [1,n)$. Erdős asked whether $\mathcal{M}(n)\le \sum_{p<n}1/p+O(1)$ uniformly in $n$. We prove the quantitative average-order formula $$ \sum_{n\le N}\mathcal{M}(n) = e^{-γ}N\log\log N+O(N). $$ The lower bound comes from the self-rough construction $\{n-d:P^{-}(n-d)>d\}$, while the upper bound uses bounded-cost dual certificates and Buchstab--de Bruijn estimates for rough numbers. We also prove that $$ \mathcal{M}(n)=(e^{-γ}+o(1))\log\log n $$ for almost all $n$, with a quantitative exceptional-set bound, and hence Erdős&#39;s inequality holds for almost all $n$. The almost-all proof uses a long-interval two-dimensional beta-sieve estimate for two moving forbidden residue classes, together with an exact finite singular-series cancellation. Finally, we prove the pointwise bound $\mathcal{M}(n)\le (2+\varepsilon)\log\log n+O_{\varepsilon}(1)$, explain the linear-sieve barrier behind the constant $2$, and record structural certificates, conditional window-packing reductions, numerical examples, and CRT sharpness constructions.