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0789dda | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | 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
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