| import Mathlib |
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| namespace OAI |
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| open Filter Asymptotics |
| open scoped Topology |
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| namespace QuarticPowerFree |
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| def PowerFree (k : ℕ) (a : ℤ) : Prop := |
| ∀ p : ℕ, p.Prime → ¬ (p : ℤ) ^ k ∣ a |
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| def rho (f : Polynomial ℤ) (q : ℕ) : ℕ := |
| ((Finset.range q).filter fun a : ℕ => (q : ℤ) ∣ f.eval (a : ℤ)).card |
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| 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 |
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| def LocallyAdmissible (f : Polynomial ℤ) (k : ℕ) : Prop := |
| ∀ p : ℕ, p.Prime → rho f (p ^ k) < p ^ k |
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| noncomputable def localFactor (f : Polynomial ℤ) (k : ℕ) (p : Nat.Primes) : ℝ := |
| 1 - (rho f (p.val ^ k) : ℝ) / (p.val : ℝ) ^ k |
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| noncomputable def densityConstant (f : Polynomial ℤ) (k : ℕ) : ℝ := |
| ∏' p : Nat.Primes, localFactor f k p |
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| 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) |
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| theorem allDegrees (f : Polynomial ℤ) |
| (hirr : Irreducible (f.map (Int.castRingHom ℚ))) |
| (hdeglo : 4 ≤ f.natDegree) |
| (hlocal : LocallyAdmissible f (f.natDegree - 2)) : |
| DensityStatement f (f.natDegree - 2) := by |
| sorry |
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| end QuarticPowerFree |
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| end OAI |
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