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Level A · Machine-checkable Hard Number theory P-erdos-975

Erdős Problem #975

For an irreducible polynomial f ∈ ℤ[x] with f(n) ≥ 1 for sufficiently large n, does there exists a constant c = c(f) > 0 such that Σ_n ≤ x τ(f(n)) ≈ c · x log x? Note that it is unclear whether the polynomial should have integer coefficients or merely be integer-valued. We assume the former.

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-975,
  title        = {Erdős Problem #975},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-975}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

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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

For an irreducible polynomial with for sufficiently large , does there exists a constant such that ?

Note that it is unclear whether the polynomial should have integer coefficients or merely be integer-valued. We assume the former.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«975». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_975 : answer(sorry) ↔
    ∀ f : ℤ[X], f.natDegree ≠ 0 → Irreducible f → (∀ᶠ n in atTop, 1 ≤ f.eval n) →
    ∃ c > (0 : ℝ), Tendsto (fun x ↦ Erdos975Sum f x / (x * log x)) atTop (𝓝 c)

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/975. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/975
  • [Va39] van der Corput, J. G., Une in\'egalit\'e{} relative au nombre des diviseurs. Nederl. Akad. Wetensch., Proc. (1939), 547--553.
  • [Er52b] Erd\"os, P., On the sum {}. J. London Math. Soc. (1952), 7--15.
  • [Ho63] Hooley, Christopher, On the number of divisors of a quadratic polynomial. Acta Math. (1963), 97--114.
  • [Mc95] McKee, James, On the average number of divisors of quadratic polynomials. Math. Proc. Cambridge Philos. Soc. (1995), 389--392.
  • [Mc97] McKee, James, A note on the number of divisors of quadratic polynomials. (1997), 275--281.
  • [Mc99] McKee, James, The average number of divisors of an irreducible quadratic polynomial. Math. Proc. Cambridge Philos. Soc. (1999), 17--22.
  • [T] T. Tao, Erdos' divisor bound, https://terrytao.wordpress.com/2011/07/23/erdos-divisor-bound/

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.