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Level A · Machine-checkable Hard Number theory P-erdos-50

Erdős Problem #50

Let f be the asymptotic distribution function of φ(n)/n, so that for each c ∈ [0,1], f(c) is the natural density of n : φ(n) < cn. Is it true that there is no x such that the derivative f'(x) exists and is positive?

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-50,
  title        = {Erdős Problem #50},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-50}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

Also: CITATION.cff · Atom feed of results

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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 be the asymptotic distribution function of , so that for each , is the natural density of . Is it true that there is no such that the derivative exists and is positive?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«50». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_50 : answer(sorry) ↔ ∀ᵉ (f : ℝ → ℝ) (hf : IsDistributionOfPhiRatio f),
    ¬∃ x ∈ Icc (0 : ℝ) 1, ∃ y > 0, HasDerivWithinAt f y (Icc 0 1) x

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/50. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/50
  • [Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry.

Resenhas (1995), 165-186.

  • [Sch38] Schoenberg, I. J. "On asymptotic distributions of arithmetical functions."

Transactions of the American Mathematical Society 39.2 (1936): 315-330.

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.