| import OAI.NumberTheory.Jacobsthal.Estimates.JointModelCount |
| import OAI.NumberTheory.Jacobsthal.Sieve.OddEulerCorrection |
|
|
| namespace OAI |
|
|
| namespace Erdos970 |
| open scoped _root_.Erdos970 |
|
|
| section |
|
|
| namespace ErdosVarianceMoments |
| open ErdosRandomVariance |
| attribute [local instance] Classical.propDecidable |
| attribute [local instance] Classical.decEq |
|
|
| theorem jointDensity_nonneg (P S : Finset ℕ) (hP : ∀ t ∈ P,t.Prime) : |
| 0 ≤ jointDensity P S := by |
| apply Finset.prod_nonneg |
| intro t ht |
| let : NeZero t := ⟨(hP t ht).ne_zero⟩ |
| exact sub_nonneg.mpr (positionDensity_bounds S t).2 |
|
|
| theorem odd_position_density_lower (P S : Finset ℕ) (hP : ∀ t ∈ P,3 ≤ t) (hS : S.card ≤ 2) : |
| (1 : ℝ)/2*(sieveProduct P)^2 ≤ jointDensity P S := by |
| have hh := ErdosOddEulerCorrection.odd_two_class_product_lower P hP |
| apply hh.trans |
| apply Finset.prod_le_prod₀ |
| · intro t ht |
| have ht3 : (3 : ℝ) ≤ t := by exact_mod_cast hP t ht |
| exact sub_nonneg.mpr ((div_le_one (by linarith)).mpr (by linarith)) |
| · intro t ht |
| let : NeZero t := ⟨by have := hP t ht;omega⟩ |
| exact sub_le_sub_left (positionDensity_le_two S hS t) 1 |
|
|
| theorem nonzero_jointDensity_lower (P S : Finset ℕ) (hP : ∀ t ∈ P,t.Prime) |
| (hS : S.card ≤ 2) (hne : jointDensity P S ≠ 0) : |
| (1 : ℝ)/2*(sieveProduct P)^2 ≤ jointDensity P S := by |
| by_cases h2 : 2 ∈ P |
| · have hodd (t : ℕ) (ht : t ∈ P.erase 2) : 3 ≤ t := by |
| have hh := Finset.mem_erase.mp ht |
| have hp := (hP t hh.2).two_le |
| omega |
| have hO := odd_position_density_lower (P.erase 2) S hodd hS |
| have hO0 := jointDensity_nonneg (P.erase 2) S (fun t ht => hP t (Finset.mem_erase.mp ht).2) |
| have hT := nonzero_positionDensity_two P S hS hne h2 |
| have hK : jointDensity P S = (1-positionDensity S 2)*jointDensity (P.erase 2) S := |
| (Finset.mul_prod_erase P (fun t => 1-positionDensity S t) h2).symm |
| have hV : sieveProduct P = (1/2 : ℝ)*sieveProduct (P.erase 2) := by |
| rw [sieveProduct,← Finset.mul_prod_erase P (fun t => 1-1/(t : ℝ)) h2] |
| norm_num [sieveProduct] |
| have hmul := mul_le_mul_of_nonneg_right (show (1 : ℝ)/2 ≤ 1-positionDensity S 2 by linarith) hO0 |
| rw [hV,hK] |
| nlinarith [sq_nonneg (sieveProduct (P.erase 2))] |
| · exact odd_position_density_lower P S (fun t ht => by |
| have hp := (hP t ht).two_le |
| have hn : t ≠ 2 := fun he => h2 (he ▸ ht) |
| omega) hS |
|
|
| end ErdosVarianceMoments |
|
|
| end |
|
|
| section |
|
|
| open _root_.Filter |
| namespace ErdosVarianceUniformMoments |
| open NumberTheoryLean ErdosVarianceMoments ErdosRandomVariance ErdosInverseEuler |
| attribute [local instance] Classical.propDecidable |
| attribute [local instance] Classical.decEq |
|
|
| noncomputable def densityInverseConstant : ℝ := 8*(Real.exp Real.eulerMascheroniConstant)^2 |
|
|
| theorem densityInverseConstant_pos : 0 < densityInverseConstant := by |
| dsimp [densityInverseConstant] |
| positivity |
|
|
| theorem actual_sieveProduct (w : ℝ) : |
| sieveProduct (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) = SmallSieveFinite.smallEuler ⌊w⌋₊ := by |
| simp only [sieveProduct,SmallSieveFinite.smallEuler,one_div] |
|
|
| theorem uniform_density_inverse : ∀ᶠ w : ℝ in atTop,2 ≤ w ∧ |
| ∀ S : Finset ℕ,S.card ≤ 2 → jointDensity (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) S ≠ 0 → |
| (jointDensity (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) S)⁻¹ ≤ |
| densityInverseConstant*(Real.log w)^2 := by |
| filter_upwards [smallEuler_log_bounds] with w hw |
| refine ⟨hw.1,?_⟩ |
| intro S hS hne |
| let P := LargePrimeDeletion.cutoffPrimes ⌊w⌋₊ |
| let V := SmallSieveFinite.smallEuler ⌊w⌋₊ |
| let K := jointDensity P S |
| let B := 2*Real.exp Real.eulerMascheroniConstant*Real.log w |
| have hP (t : ℕ) (ht : t ∈ P) : t.Prime := (LargePrimeDeletion.mem_cutoffPrimes.mp ht).1 |
| have hlow := nonzero_jointDensity_lower P S hP hS hne |
| rw [actual_sieveProduct] at hlow |
| have hV : 0 < V := hw.2.1 |
| have hVB : 1 ≤ V*B := by |
| have hh := mul_le_mul_of_nonneg_left hw.2.2.2 hV.le |
| change V*V⁻¹ ≤ V*B at hh |
| simpa only [mul_inv_cancel₀ hV.ne'] using hh |
| have hsq : (1 : ℝ) ≤ V^2*B^2 := by |
| have hh := pow_le_pow_left₀ (by norm_num : (0 : ℝ) ≤ 1) hVB 2 |
| simpa only [one_pow,mul_pow] using hh |
| have hm := mul_le_mul_of_nonneg_right hlow (sq_nonneg B) |
| have hKpos : 0 < K := (by dsimp [K,V] at *;positivity : 0 < (1 : ℝ)/2*V^2).trans_le hlow |
| have hb : 2*B^2 = densityInverseConstant*(Real.log w)^2 := by |
| dsimp [B,densityInverseConstant] |
| ring |
| have hmain : 1 ≤ (densityInverseConstant*(Real.log w)^2)*K := by |
| rw [← hb] |
| dsimp [K,V] at * |
| nlinarith |
| rw [← one_div] |
| exact (div_le_iff₀ hKpos).mpr hmain |
| |
| end ErdosVarianceUniformMoments |
| |
| end |
| |
| section |
| |
| namespace ErdosVarianceMoments |
| open NumberTheoryLean ErdosVarianceSmallModel |
| attribute [local instance] Classical.propDecidable |
| attribute [local instance] Classical.decEq |
| |
| theorem jointDensity_le_one (P S : Finset ℕ) (hP : ∀ t ∈ P,t.Prime) : jointDensity P S ≤ 1 := by |
| apply Finset.prod_le_one₀ |
| · intro t ht |
| let : NeZero t := ⟨(hP t ht).ne_zero⟩ |
| exact sub_nonneg.mpr (positionDensity_bounds S t).2 |
| · intro t ht |
| let : NeZero t := ⟨(hP t ht).ne_zero⟩ |
| have hh := (positionDensity_bounds S t).1 |
| linarith |
| |
| theorem jointDensity_split (w : ℝ) (H : ℕ) (S : Finset ℕ) : |
| jointDensity (divisorPrimes w H) S*jointDensity (coprimePrimes w H) S = |
| jointDensity (LargePrimeDeletion.cutoffPrimes ⌊w⌋₊) S := by |
| unfold jointDensity divisorPrimes coprimePrimes |
| exact Finset.prod_filter_mul_prod_filter_not _ _ _ |
| |
| theorem model_normalization_identity (R xi w : ℝ) (H : ℕ) (C0 : ℤ) |
| (hR : 0 < R) (hxi : 0 ≤ xi) (hH : 0 < H) : |
| atomWeight R xi w H C0*(divisorModulus w H : ℝ)* |
| ((coprimeModulus w H).totient : ℝ)* |
| (modelLength R xi (H : ℝ) (C0 : ℝ)*((H.totient : ℝ)/(H : ℝ))) = 1 := by |
| have hHN : (H : ℝ) ≠ 0 := by exact_mod_cast hH.ne' |
| have hphi : (H.totient : ℝ) ≠ 0 := by exact_mod_cast (Nat.totient_pos.mpr hH).ne' |
| have hT0 : (divisorModulus w H : ℝ) ≠ 0 := by exact_mod_cast (divisorModulus_pos w H).ne' |
| have hT1 : ((coprimeModulus w H).totient : ℝ) ≠ 0 := by |
| exact_mod_cast (Nat.totient_pos.mpr (coprimeModulus_pos w H)).ne' |
| have hY : 0 < modelLength R xi (H : ℝ) (C0 : ℝ) := by |
| have hHR : (0 : ℝ) < H := by exact_mod_cast hH |
| dsimp [modelLength,interceptScale] |
| positivity |
| unfold atomWeight |
| field_simp |
| |
| end ErdosVarianceMoments |
| |
| end |
| |
| end Erdos970 |
| |
| end OAI |
| |
