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8769036 | 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 | import OAI.NumberTheory.TwoPoint.Bounds.ConditionalLiouville
import OAI.NumberTheory.TwoPoint.Bounds.ConditionalElliott
namespace OAI
/-! Paper 009-01, all three main results, with their published inputs explicit. -/
namespace OrdinaryTwoPointCorrelationsConditional
open TwoPointCorrelations
open Filter
theorem liouville_log_saving (hP : ModFiveThetaInput)
(hBr : BravermanDepth22Input) (hM : PrimeReciprocalInput)
(hMRT : MRTLiouvilleShortInput) :
∃ 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 :=
conditional_liouvilleLogSaving hP hBr hM hMRT
theorem binary_corrected_elliott (hP : ModFiveThetaInput)
(hBr : BravermanDepth22Input) (hM : PrimeReciprocalInput)
(hMRT : MRTShortExponentialInput) :
∀ 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) :=
conditional_binaryCorrectedElliott hP hBr hM hMRT
theorem affine_corrected_elliott (hP : ModFiveThetaInput)
(hBr : BravermanDepth22Input) (hM : PrimeReciprocalInput)
(hMRT : MRTShortExponentialInput) :
∀ 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) :=
conditional_affineCorrectedElliott hP hBr hM hMRT
end OrdinaryTwoPointCorrelationsConditional
end OAI
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