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

Abstract

For $n\ge 2$, let $M(n)$ be the supremum of $\sum_{a\in A}1/(n-a)$ over pairwise coprime sets $A\subset[1,n)$. Erd\H{o}s asked whether $M(n)\le \sum_{p d\}$ with a bounded-cost dual certificate and Buchstab--de Bruijn estimates. We also show $M(n)=(e^{-\gamma}+o(1))\log\log n$ for almost all $n$, with a quantitative exceptional-set bound, so Erd\H{o}s's inequality holds for almost all $n$. The argument combines a long-interval two-dimensional beta sieve for two moving forbidden residue classes with an exact finite singular-series cancellation. The remaining obstacles to a full variance estimate are a shifted-prime second moment for $d>\sqrt{2N}$ and finite-$u$ rough correlations in the intermediate tail, yielding a precise conditional criterion. For every $\varepsilon>0$ we also prove the uniform pointwise bound $M(n)\le(2+\varepsilon)\log\log n+O_{\varepsilon}(1)$ and explain the linear-sieve barrier at the constant $2$. The results of this paper have been formally verified in Lean.