| import Mathlib | |
| namespace OAI | |
| open Filter Asymptotics | |
| open scoped Topology | |
| namespace QuarticPowerFree | |
| def PowerFree (k : ℕ) (a : ℤ) : Prop := | |
| ∀ p : ℕ, p.Prime → ¬ (p : ℤ) ^ k ∣ a | |
| def rho (f : Polynomial ℤ) (q : ℕ) : ℕ := | |
| ((Finset.range q).filter fun a : ℕ => (q : ℤ) ∣ f.eval (a : ℤ)).card | |
| noncomputable def count (f : Polynomial ℤ) (k : ℕ) (X : ℝ) : ℕ := by | |
| classical | |
| exact ((Finset.Icc 1 ⌊X⌋₊).filter fun n : ℕ => PowerFree k (f.eval (n : ℤ))).card | |
| def LocallyAdmissible (f : Polynomial ℤ) (k : ℕ) : Prop := | |
| ∀ p : ℕ, p.Prime → rho f (p ^ k) < p ^ k | |
| noncomputable def localFactor (f : Polynomial ℤ) (k : ℕ) (p : Nat.Primes) : ℝ := | |
| 1 - (rho f (p.val ^ k) : ℝ) / (p.val : ℝ) ^ k | |
| noncomputable def densityConstant (f : Polynomial ℤ) (k : ℕ) : ℝ := | |
| ∏' p : Nat.Primes, localFactor f k p | |
| def DensityStatement (f : Polynomial ℤ) (k : ℕ) : Prop := | |
| Multipliable (localFactor f k) ∧ | |
| 0 < densityConstant f k ∧ | |
| (fun X : ℝ => (count f k X : ℝ) - densityConstant f k * X) =o[atTop] | |
| (fun X : ℝ => X) | |
| end QuarticPowerFree | |
| end OAI | |