Erdős (Erdos) Problem #975: status in 2026

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.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #975 as open (status updated 31 August 2025).
Formal Conjectures marks the Lean statement erdos_975 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #975 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: status of `erdos_975`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 5 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

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 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

For an irreducible polynomial $f \in Z [x]$ with $f (n) \geq 1$ for sufficiently large $n$ , does there exists a constant $c = c (f) > 0$ such that $\sum_{n \leq x} τ (f (n)) \approx c \cdot x lo g x$ ?

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».

theorem erdos_975 :
    ∀ 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 { $\sum_{k = 1}^{x} d (f (k))$ }. 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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.