| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators ComplexConjugate |
| open Filter |
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| namespace TwoPointCorrelations |
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| def liouville (n : ℕ) : ℂ := ArithmeticFunction.liouville n |
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| def correlationSum (f g : ℕ → ℂ) (h₁ h₂ N : ℕ) : ℂ := |
| ∑ n ∈ Finset.Icc 1 N, f (n + h₁) * g (n + h₂) |
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| def affineSum (f g : ℕ → ℂ) (a₁ a₂ b₁ b₂ N : ℕ) : ℂ := |
| ∑ n ∈ Finset.Icc 1 N, f (a₁ * n + b₁) * g (a₂ * n + b₂) |
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| def Multiplicative (f : ℕ → ℂ) : Prop := |
| ∀ m n : ℕ, 0 < m → 0 < n → m.Coprime n → f (m * n) = f m * f n |
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| def OneBounded (f : ℕ → ℂ) : Prop := ∀ n : ℕ, 0 < n → ‖f n‖ ≤ 1 |
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| def primesUpTo (N : ℕ) : Finset ℕ := (Finset.range (N + 1)).filter Nat.Prime |
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| def squaredDistance (f g : ℕ → ℂ) (N : ℕ) : ℝ := |
| ∑ p ∈ primesUpTo N, (1 - (f p * conj (g p)).re) / (p : ℝ) |
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| def characterTwist {q : ℕ} (χ : DirichletCharacter ℂ q) |
| (t : ℝ) (n : ℕ) : ℂ := |
| χ (n : ZMod q) * Complex.exp (((t * Real.log (n : ℝ) : ℝ) : ℂ) * Complex.I) |
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| def UniformlyNonpretentious (f : ℕ → ℂ) : Prop := |
| ∀ (q : ℕ), 0 < q → ∀ (χ : DirichletCharacter ℂ q) (K : ℝ), |
| ∀ᶠ N : ℕ in atTop, ∀ t : ℝ, |t| ≤ (N : ℝ) → |
| K ≤ squaredDistance f (characterTwist χ t) N |
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| end TwoPointCorrelations |
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| namespace OrdinaryTwoPointCorrelations |
| open TwoPointCorrelations |
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| 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 := by |
| sorry |
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| 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) := by |
| sorry |
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| 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) := by |
| sorry |
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| end OrdinaryTwoPointCorrelations |
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| end |
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| end OAI |
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|
