False ↔ ∀ (f : ℕ → ℝ), (∀ a > 0, ∀ b > 0, a.Coprime b → f (a * b) = f a + f b) → Filter.limsup (fun x => match x with | (p, k) => ↑(f (p ^ k)) / ↑(Real.log (↑p ^ k))) (Filter.atTop ⊓ Filter.principal {(p, k) | Nat.Prime p}) = ⊤ → Filter.limsup (fun n => (↑(f (n + 1)) - ↑(f n)) / ↑(Real.log ↑n)) Filter.atTop = ⊤