import OAI.NumberTheory.TwoPoint.Main

namespace OAI

namespace OrdinaryTwoPointCorrelations

open TwoPointCorrelations
open Filter

theorem liouville_log_saving :
    ∃ 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 :=
  TwoPointCorrelations.liouvilleLogSaving

theorem binary_corrected_elliott :
    ∀ 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) :=
  TwoPointCorrelations.binaryCorrectedElliott

theorem affine_corrected_elliott :
    ∀ 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) :=
  TwoPointCorrelations.affineCorrectedElliott

end OrdinaryTwoPointCorrelations

end OAI