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

Mathematics > Number Theory [Submitted on 16 Jun 2026] Title:An Average-Order Theorem for a Shifted Pairwise-Coprime Extremal Problem View PDF HTML (experimental)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 $$ \sum_{n\le N}\mathcal{M}(n) = e^{-\gamma}N\log\log N+O(N). $$ 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 $$ \mathcal{M}(n)=(e^{-\gamma}+o(1))\log\log n $$ 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 $\mathcal{M}(n)\le (2+\varepsilon)\log\log n+O_{\varepsilon}(1)$, explain the linear-sieve barrier behind the constant $2$, and record structural certificates, conditional window-packing reductions, numerical examples, and CRT sharpness constructions. Current browse context: math.NT References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.