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180bc03 | 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 | import OAI.NumberTheory.TwoPoint.Basic
import OAI.NumberTheory.TwoPoint.PretentiousDistance
import Mathlib.Analysis.SpecialFunctions.Pow.Real
namespace OAI
open Filter
namespace TwoPointCorrelations
/-- Theorem 1.1 (`thm:q-affine`) of the manuscript, through all real
cutoffs, with a single exponent independent of the four affine coefficients. -/
def LiouvilleLogSaving : Prop :=
∃ 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
/-- Theorem 1.2 (`thm:elliott`): the factors are ordinary multiplicative
functions; either original factor satisfies the growing-height condition. -/
def BinaryCorrectedElliott : Prop :=
∀ 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)
/-- Corollary 1.3 (`cor:affine`), including coefficients sharing prime factors. -/
def AffineCorrectedElliott : Prop :=
∀ 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)
end TwoPointCorrelations
end OAI
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