Erdős Problem 277

References:

• erdosproblems.com/277
• [Ha79] Haight, J. A., Covering systems of congruences, a negative result. Mathematika (1979), 53--61.

theorem Erdos277.erdos_277 : True ↔ ∀ (c : ℝ), ∃ (n : ℕ), ↑((ArithmeticFunction.sigma 1) n) > c * ↑n ∧ ¬∃ (S : Finset (ℤ × ℕ)), (∀ (z : ℤ), ∃ s ∈ S, z ≡ s.1 [ZMOD ↑s.2]) ∧ ∀ s ∈ S, s.2 ∣ n ∧ 1 < s.2

Is it true that, for every $c$, there exists an $n$ such that $\sigma(n)>cn$ but there is no covering system whose moduli all divide $n$?

This was answered affirmatively by Haight [Ha79].