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Erdos's Question on Shifted Pairwise-Coprime Reciprocal Sums

Let M(n)\mathcal{M}(n) be the supremum of ∑a∈A1/(n−a)\sum_{a \in A} 1/(n-a) over pairwise coprime A⊂[1,n)A \subset [1,n). Erdos asked whether M(n)≤∑p<n1/p+O(1)\mathcal{M}(n) \le \sum_{p<n} 1/p + O(1) uniformly. The average order is settled: ∑n≤NM(n)=e−γNlog⁡log⁡N+O(N)\sum_{n \le N} \mathcal{M}(n) = e^{-\gamma} N \log\log N + O(N).

Result
Proved(see note)
Status
Partial result
AI contribution
AI co-developed
Method
Argument
Field
Number theory
Posed by
Paul Erdos
Year posed
1980
Years open
46y
Solved
2026-06-16
Model
ChatGPT
Vendor
OpenAI
Collaborators
Eric Li
Verification
Unreviewed
Publication
Preprint
Significance
10 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

the average order; the uniform bound Erdos asked about is not settled

What the AI did

The declaration says ChatGPT was used for ideation and formalization during preparation, with the author responsible for the mathematics. Part of the same series of Erdos-problem resolutions in this catalog.

Verification

Single-author arXiv preprint; not yet peer-reviewed.

Sources

Changelog1 change
  • Matarisvanset Significance note to “A question of Erdos on pairwise coprime sets, documented and narrow.”, also Year posed, More links, Age footnote, Significance note

Discussion