What was already known
Erdős and Hall asked for this fixed-scale limit. Ford bounded the corresponding interval count up to constants, but that did not determine the limiting ratio.
AI involvement
OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.
OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.
Importance
Answers the Erdős–Hall fixed-scale question and accompanies an explicit asymptotic equivalent for V(x).
Limitations
The claim concerns fixed c, not a uniform limit as c varies. The pinned Lean scope note reports a check of the explicit asymptotic and fixed-scale ratio; it was not rerun here.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean scope note includes the explicit asymptotic and the fixed-scale limit V(cx)/V(x)→c.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- If V(x) counts distinct values of Euler’s totient function up to x, then V(cx)/V(x) tends to c for every fixed real c>0.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.
Correction and publication history
Primary manuscript, attribution, prior-work, and scoped checking review for non-notifying local inclusion.
Read this published revision