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

For $n\ge 2$, let $\mathcal{M}(n)$ be the supremum of ${\sum }_{a\in A}1\mathrm{/}(n-a)$ over pairwise coprime sets $A\subset [1,n)$. Erdős asked whether $\mathcal{M}(n)\le {\sum }_{p<n}1\mathrm{/}p+O(1)$ uniformly in $n$. We prove the quantitative average-order formula $$\sum _{n\le N}\mathcal{M}(n)=e^{-\gamma }N\log \log N+O(N)\mathrm{.}$$ 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 $$\mathcal{M}(n)=(e^{-\gamma }+o(1))\log \log n$$ for almost all $n$, with a quantitative exceptional-set bound, and hence Erdős’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 $\mathcal{M}(n)\le (2+\epsilon )\log \log n+O_{\epsilon }(1)$, explain the linear-sieve barrier behind the constant $2$, and record structural certificates, conditional window-packing reductions, numerical examples, and CRT sharpness constructions.