File size: 1,380 Bytes
465b10d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 | 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)
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
end QuarticPowerFree
end OAI
|