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Level A · Machine-checkable Hard Analysis P-erdos-513

Erdős Problem #513

Let f be a transcendental entire function. What is the greatest possible value of liminf (fun r : ℝ => ratio r f) atTop?

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-513,
  title        = {Erdős Problem #513},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-513}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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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

Let f be a transcendental entire function. What is the greatest possible value of liminf (fun r : ℝ => ratio r f) atTop?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«513». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_513 : answer(sorry) =
    ⨆ f : {f : ℂ → ℂ // Transcendental ℂ[X] f ∧ Differentiable ℂ f},
    (liminf (fun r : ℝ => ratio r f) atTop)

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/513. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/513
  • [ClHa64] Clunie, J. and Hayman, W. K., The maximum term of a power series. J. Analyse Math. (1964), 143-186.

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.