import Mathlib

namespace OAI

noncomputable section
open scoped BigOperators
open Filter

namespace OrdinaryCorrelations

def OneBounded (f : ℕ → ℂ) : Prop := ∀ n, ‖f n‖ ≤ 1

def Multiplicative (f : ℕ → ℂ) : Prop :=
  ∀ m n : ℕ, 0 < m → 0 < n → m.Coprime n → f (m * n) = f m * f n

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)

def distance (f : ℕ → ℂ) {q : ℕ} (χ : DirichletCharacter ℂ q)
    (t X : ℝ) : ℝ := Real.sqrt (distanceSq f χ t X)

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

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)

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

end OAI