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

[2606.17955] An Average-Order Theorem for a Shifted Pairwise-Coprime Extremal Problem Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2606.17955 (math) [Submitted on 16 Jun 2026 ( v1 ), last revised 15 Aug 2026 (this version, v2)] Title: An Average-Order Theorem for a Shifted Pairwise-Coprime Extremal Problem Authors: Eric Li (University of Cambridge) View a PDF of the paper titled An Average-Order Theorem for a Shifted Pairwise-Coprime Extremal Problem, by Eric Li (University of Cambridge) View PDF HTML (experimental) Abstract: For $n\ge 2$, let $M(n)$ be the supremum of $\sum_{a\in A}1/(n-a)$ over pairwise coprime sets $A\subset[1,n)$. Erdős asked whether $M(n)\le \sum_{p<n}1/p+O(1)$ uniformly in $n$. We prove the quantitative average-order formula $\sum_{n\le N}M(n)=e^{-\gamma}N\log\log N+O(N)$. The proof combines the self-rough lower construction $\{n-d:P^{-}(n-d)>d\}$ 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 Erdős'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. Comments: 36 pages, no figures. v2 adds an accompanying Lean 4 formalisation, a formalisation section and repository link, and expanded acknowledgements; exposition revised Subjects: Number Theory (math.NT) ; Combinatorics (math.CO) MSC classes: 11N25 (Primary), 11A05, 11B05, 11N36 (Secondary) Cite as: arXiv:2606.17955 [math.NT] (or arXiv:2606.17955v2 [math.NT] for this version) https://doi.org/10.48550/arXiv.2606.17955 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Eric Li [ view email ] [v1] Tue, 16 Jun 2026 14:08:32 UTC (33 KB) [v2] Sat, 15 Aug 2026 03:02:38 UTC (34 KB) Full-text links: Access Paper: View a PDF of the paper titled An Average-Order Theorem for a Shifted Pairwise-Coprime Extremal Problem, by Eric Li (University of Cambridge) View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2026-06 Change to browse by: math math.CO References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs 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 . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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