Erdős Problem #538 — open problem in number theory

Erdős Problem #538

Let r≥ 2 and suppose that A⊆1,…,N is such that, for any m, there are at most r solutions to m=pa where p is prime and a∈ A. Give the best possible upper bound for Σ_n∈ A1/n. Erdős observed that Σ_n∈ A1/n≪ rlog N/loglog N, and the order Θ_r(log N / loglog N) is known (see erdos_538.matching_order).

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-538,
  title        = {Erdős Problem #538},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-538}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $r \geq 2$ and suppose that $A \subseteq {1, \dots, N}$ is such that, for any $m$ , there are at most $r$ solutions to $m = p a$ where $p$ is prime and $a \in A$ . Give the best possible upper bound for $\sum_{n \in A} \frac{1}{n}$ .

Erdős observed that $\sum_{n \in A} \frac{1}{n} ≪ r \frac{l o g N}{l o g l o g N}$ , and the order Θ_r(log N / loglog N) is known (see erdos_538.matching_order). The best possible upper bound is the asymptotic size of the largest reciprocal sum maxMass r N over admissible A.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«538». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_538 :
    let f : ℕ → ℕ → ℝ := answer(sorry)
    ∀ r : ℕ, 2 ≤ r → maxMass r ~[atTop] f 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/538. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/538

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.