| import OAI.NumberTheory.TwoPoint.Bounds.ConditionalLiouville |
| import OAI.NumberTheory.TwoPoint.Bounds.ConditionalElliott |
|
|
| namespace OAI |
|
|
| /-! Paper 009-01, all three main results, with their published inputs explicit. -/ |
|
|
| namespace OrdinaryTwoPointCorrelationsConditional |
|
|
| open TwoPointCorrelations |
| open Filter |
|
|
| theorem liouville_log_saving (hP : ModFiveThetaInput) |
| (hBr : BravermanDepth22Input) (hM : PrimeReciprocalInput) |
| (hMRT : MRTLiouvilleShortInput) : |
| ∃ c : ℝ, 0 < c ∧ ∀ a₁ a₂ b₁ b₂ : ℕ, |
| 0 < a₁ → 0 < a₂ → a₁ * b₂ ≠ a₂ * b₁ → |
| ∃ C : ℝ, 0 < C ∧ ∀ X : ℝ, 3 ≤ X → |
| ‖affineSum liouville liouville a₁ a₂ b₁ b₂ ⌊X⌋₊‖ ≤ |
| C * X / Real.rpow (Real.log X) c := |
| conditional_liouvilleLogSaving hP hBr hM hMRT |
|
|
| theorem binary_corrected_elliott (hP : ModFiveThetaInput) |
| (hBr : BravermanDepth22Input) (hM : PrimeReciprocalInput) |
| (hMRT : MRTShortExponentialInput) : |
| ∀ f₁ f₂ : ℕ → ℂ, Multiplicative f₁ → Multiplicative f₂ → |
| OneBounded f₁ → OneBounded f₂ → |
| (UniformlyNonpretentious f₁ ∨ UniformlyNonpretentious f₂) → |
| ∀ h₁ h₂ : ℕ, h₁ ≠ h₂ → |
| Tendsto (fun N : ℕ => correlationSum f₁ f₂ h₁ h₂ N / (N : ℂ)) atTop (nhds 0) := |
| conditional_binaryCorrectedElliott hP hBr hM hMRT |
|
|
| theorem affine_corrected_elliott (hP : ModFiveThetaInput) |
| (hBr : BravermanDepth22Input) (hM : PrimeReciprocalInput) |
| (hMRT : MRTShortExponentialInput) : |
| ∀ f₁ f₂ : ℕ → ℂ, Multiplicative f₁ → Multiplicative f₂ → |
| OneBounded f₁ → OneBounded f₂ → |
| (UniformlyNonpretentious f₁ ∨ UniformlyNonpretentious f₂) → |
| ∀ a₁ a₂ b₁ b₂ : ℕ, 0 < a₁ → 0 < a₂ → a₁ * b₂ ≠ a₂ * b₁ → |
| Tendsto (fun N : ℕ => affineSum f₁ f₂ a₁ a₂ b₁ b₂ N / (N : ℂ)) atTop (nhds 0) := |
| conditional_affineCorrectedElliott hP hBr hM hMRT |
|
|
| end OrdinaryTwoPointCorrelationsConditional |
|
|
| end OAI |
|
|
