File size: 2,641 Bytes
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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 | import Mathlib
namespace OAI
noncomputable section
open scoped BigOperators ComplexConjugate
open Filter
namespace TwoPointCorrelations
def liouville (n : ℕ) : ℂ := ArithmeticFunction.liouville n
def correlationSum (f g : ℕ → ℂ) (h₁ h₂ N : ℕ) : ℂ :=
∑ n ∈ Finset.Icc 1 N, f (n + h₁) * g (n + h₂)
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)
def Multiplicative (f : ℕ → ℂ) : Prop :=
∀ m n : ℕ, 0 < m → 0 < n → m.Coprime n → f (m * n) = f m * f n
def OneBounded (f : ℕ → ℂ) : Prop := ∀ n : ℕ, 0 < n → ‖f n‖ ≤ 1
def primesUpTo (N : ℕ) : Finset ℕ := (Finset.range (N + 1)).filter Nat.Prime
def squaredDistance (f g : ℕ → ℂ) (N : ℕ) : ℝ :=
∑ p ∈ primesUpTo N, (1 - (f p * conj (g p)).re) / (p : ℝ)
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)
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
end TwoPointCorrelations
namespace OrdinaryTwoPointCorrelations
open TwoPointCorrelations
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
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
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
end OrdinaryTwoPointCorrelations
end
end OAI
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