| import Mathlib |
|
|
| namespace OAI |
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|
| noncomputable section |
| open scoped BigOperators |
| open Filter |
|
|
| namespace OrdinaryCorrelations |
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|
| def OneBounded (f : ℕ → ℂ) : Prop := ∀ n, ‖f n‖ ≤ 1 |
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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 distanceSq (f : ℕ → ℂ) {q : ℕ} (χ : DirichletCharacter ℂ q) |
| (t X : ℝ) : ℝ := |
| ∑ p ∈ (Finset.Icc 2 ⌊X⌋₊).filter Nat.Prime, |
| (1 - (f p * star (χ (p : ZMod q) * |
| Complex.exp ((t * Real.log (p : ℝ) : ℝ) * Complex.I))).re) / (p : ℝ) |
|
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| def distance (f : ℕ → ℂ) {q : ℕ} (χ : DirichletCharacter ℂ q) |
| (t X : ℝ) : ℝ := Real.sqrt (distanceSq f χ t X) |
|
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| def UniformlyNonpretentious (f : ℕ → ℂ) : Prop := |
| ∀ (q : ℕ), 0 < q → ∀ χ : DirichletCharacter ℂ q, |
| Tendsto (fun N : ℕ => sInf ((fun t : ℝ => distance f χ t (N : ℝ)) '' |
| Set.Icc (-(N : ℝ)) (N : ℝ))) atTop atTop |
|
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| def shiftAverage (f₁ f₂ : ℕ → ℂ) (h₁ h₂ N : ℕ) : ℂ := |
| (N : ℂ)⁻¹ * ∑ n ∈ Finset.Icc 1 N, f₁ (n + h₁) * f₂ (n + h₂) |
|
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| theorem binary_corrected_elliott |
| (f₁ f₂ : ℕ → ℂ) (hf₁ : OneBounded f₁) (hf₂ : OneBounded f₂) |
| (hm₁ : Multiplicative f₁) (hm₂ : Multiplicative f₂) |
| (hNP : UniformlyNonpretentious f₁ ∨ UniformlyNonpretentious f₂) |
| (h₁ h₂ : ℕ) (hne : h₁ ≠ h₂) : |
| Tendsto (shiftAverage f₁ f₂ h₁ h₂) atTop (nhds 0) := by |
| sorry |
|
|
| end OrdinaryCorrelations |
| end |
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| end OAI |
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|
