| import OAI.NumberTheory.Jacobsthal.Sieve.IntegerLaw |
|
|
| namespace OAI |
|
|
| namespace Erdos970 |
|
|
|
|
| namespace EulerWeightedLower |
| open Erdos970.EulerPrimeLaw |
| attribute [local instance] Classical.propDecidable |
|
|
| noncomputable def lowerInteger (E : Finset ℕ) (s : Outcomes E) : ℕ := |
| integerProduct E (fun p => lowerExponent (s p)) |
| noncomputable def upperInteger (E : Finset ℕ) (s : Outcomes E) : ℕ := |
| integerProduct E (fun p => upperExponent (s p)) |
|
|
| variable (E : Finset ℕ) (hE : ∀ p ∈ E, Nat.Prime p) |
| include hE |
|
|
| lemma log_integerProduct (k : E → ℕ) : |
| Real.log (integerProduct E k : ℝ) = ∑ p : E, (k p : ℝ) * Real.log (p : ℝ) := by |
| rw [integerProduct, Nat.cast_prod] |
| rw [Real.log_prod] |
| · simp only [Nat.cast_pow, Real.log_pow] |
| · intro p _; exact_mod_cast pow_ne_zero (k p) (hE p p.property).ne_zero |
|
|
| theorem upper_log_eq_lower_add_excess (s : Outcomes E) : |
| Real.log (upperInteger E s : ℝ) = Real.log (lowerInteger E s : ℝ) + excessLog E s := by |
| rw [upperInteger, lowerInteger, log_integerProduct E hE, log_integerProduct E hE] |
| unfold excessLog |
| rw [← Finset.sum_add_distrib] |
| apply Finset.sum_congr rfl |
| intro p _ |
| rw [Nat.cast_sub (lowerExponent_le_upperExponent (s p))] |
| ring |
|
|
| theorem source_cutoff_implication {P : ℝ} (hP : 1 < P) (s : Outcomes E) |
| (hl : (lowerInteger E s : ℝ) ≤ P ^ (94 / 100 : ℝ)) |
| (hz : excessLog E s ≤ (1 / 100 : ℝ) * Real.log P) : |
| (upperInteger E s : ℝ) ≤ P ^ (95 / 100 : ℝ) := by |
| have hu : 0 < (upperInteger E s : ℝ) := by |
| exact_mod_cast integerProduct_pos E hE (fun p => upperExponent (s p)) |
| have hl0 : 0 < (lowerInteger E s : ℝ) := by |
| exact_mod_cast integerProduct_pos E hE (fun p => lowerExponent (s p)) |
| apply (Real.log_le_log_iff hu (Real.rpow_pos_of_pos (by linarith) _)).mp |
| rw [upper_log_eq_lower_add_excess E hE] |
| have hh := Real.log_le_log hl0 hl |
| rw [Real.log_rpow (by linarith : 0 < P)] at hh ⊢ |
| linarith |
|
|
| theorem small_product_coupling_lower {P : ℝ} (hP : 1 < P) : |
| eventMass E (fun s => (lowerInteger E s : ℝ) ≤ P ^ (94 / 100 : ℝ)) - |
| 100 * uniformMomentBound / Real.log P ≤ |
| eventMass E (fun s => (upperInteger E s : ℝ) ≤ P ^ (95 / 100 : ℝ)) := by |
| have htwo : ∀ p ∈ E, 2 ≤ p := fun p hp => (hE p hp).two_le |
| let L : Outcomes E → Prop := fun s => (lowerInteger E s : ℝ) ≤ P ^ (94 / 100 : ℝ) |
| let U : Outcomes E → Prop := fun s => (upperInteger E s : ℝ) ≤ P ^ (95 / 100 : ℝ) |
| let B : Outcomes E → Prop := fun s => (1 / 100 : ℝ) * Real.log P < excessLog E s |
| have hi : eventMass E L ≤ eventMass E U + eventMass E B := by |
| rw [eventMass, eventMass, eventMass, ← Summable.tsum_add |
| (product_event_summable E htwo U) (product_event_summable E htwo B)] |
| apply Summable.tsum_le_tsum _ (product_event_summable E htwo L) |
| ((product_event_summable E htwo U).add (product_event_summable E htwo B)) |
| intro s |
| have hw := productMass_nonneg E htwo s |
| by_cases hl : L s |
| · by_cases hz : B s |
| · simp only [ite_eq_left hl, ite_eq_left hz]; split_ifs <;> linarith |
| · have hu : U s := source_cutoff_implication E hE hP s hl (le_of_not_gt hz) |
| simp only [ite_eq_left hl, ite_eq_left hu, ite_eq_right hz, add_zero, le_refl] |
| · simp only [ite_eq_right hl]; split_ifs <;> linarith |
| have ht := excess_tail_le E htwo (show 0 < (1 / 100 : ℝ) * Real.log P by |
| exact mul_pos (by norm_num) (Real.log_pos hP)) |
| have he : uniformMomentBound / ((1 / 100 : ℝ) * Real.log P) = |
| 100 * uniformMomentBound / Real.log P := by ring |
| rw [he] at ht |
| change eventMass E L - _ ≤ eventMass E U |
| change eventMass E B ≤ _ at ht |
| linarith |
|
|
| end EulerWeightedLower |
|
|
|
|
| end Erdos970 |
|
|
| end OAI |
|
|
