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Level A · Machine-checkable Hard Number theory P-erdos-912

Erdős Problem #912

Prove that there exists some c>0 such that h(n) ∼ c (n/log n)^1/2 as n→ ∞.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite 
@misc{cairn-erdos-912,
  title        = {Erdős Problem #912},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-912}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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On the literature board
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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Prove that there exists some such that as .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«912».

theorem erdos_912 : ∃ c > 0,
    (fun n => (h n : ℝ)) ~[atTop] (fun n => c * (n / Real.log n) ^ (1 / 2 : ℝ))

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already — check erdosproblems.com/912. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_912.variants.tao — A heuristic of Tao using the Cramér model for the primes suggests this is true with c=√(2π).

References

  • erdosproblems.com/912
  • [Er82c] Erdős, P., Miscellaneous problems in number theory. Congr. Numer. (1982), 25-45.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.