Erdős Problem 1210

References:

• erdosproblems.com/1210
• [Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
• [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.

theorem Erdos1210.erdos_1210 : True ↔ ∃ (C : ℝ), ∀ (n : ℕ) (A : Finset ℕ), (∀ a ∈ A, 1 ≤ a ∧ a < n) → (∀ a ∈ A, ∀ b ∈ A, a ≠ b → a.Coprime b) → ∑ a ∈ A, 1 / (↑n - ↑a) ≤ ∑ p ∈ Finset.range n with Prime p, 1 / ↑p + C

Let $A\subseteq [1,n)$ be a set of integers such that $(a,b)=1$ for all distinct $a,b\in A$. Is it true that $\sum_{a\in A}\frac{1}{n-a}\leq \sum_{p < n}\frac{1}{p}+O(1)$?

theorem Erdos1210.erdos_1210.variants.er80_correction : True ↔ ∃ (C : ℝ), ∀ (n m : ℕ), n < m → ∑ q ∈ Finset.Ioc n m with Prime q, 1 / (↑q - ↑n) < ∑ p ∈ Finset.range (m - n) with Prime p, 1 / ↑p + C

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 $n < q_1 < \cdots < q_k\leq m$ is the set of primes in $(n,m]$ then $\sum \frac{1}{q_i-n} < \sum_{p < m-n}\frac{1}{p}+O(1)$.