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OP-9411OpenNumber theory

Erdős #1210 · number theory

Let A⊆[1,n)A\subseteq [1,n) be a set of integers such that (a,b)=1(a,b)=1 for all distinct a,b∈Aa,b\in A. Is it true that

∑a∈A1n−a≤∑p<n1p+O(1)?\sum_{a\in A}\frac{1}{n-a}\leq \sum_{p<n}\frac{1}{p}+O(1)?
MSC 11 · Number theory

Inferred from the community field; finer classification has not been reviewed.

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let A⊆[1,n)A\subseteq [1,n) be a set of integers such that (a,b)=1(a,b)=1 for all distinct a,b∈Aa,b\in A. Is it true that

∑a∈A1n−a≤∑p<n1p+O(1)?\sum_{a\in A}\frac{1}{n-a}\leq \sum_{p<n}\frac{1}{p}+O(1)?

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Formal statement

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Background

In [Er80] he claims he 'did not state [this] quite correctly' in [Er77c]. The problem in [Er77c] which Erdős is presumably referring to states that if n<q1<⋯<qk≤mn<q_1<\cdots<q_k\leq m is the set of primes in (n,m](n,m] then

∑1qi−n<∑p<m−n1p+O(1).\sum \frac{1}{q_i-n}< \sum_{p<m-n}\frac{1}{p}+O(1).

See also [460] and [950].

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1210 link
  2. [2]Upstream Lean 4 statement for Erdős #1210 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 1210 link

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