QualitativeStrongRoughShifts.lean`

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The stronger qualitative rough-shift estimate on exp(C B²) scales, uniform over finite-prime modifications, progression classes, and bounded rough coefficients.

theorem OAI.TwoPointCorrelations.qualitative_strong_rough_shifts line 20

Source documentation

The rough analytic transfer for a nonpretentious first factor. All parameters other than the long average length are fixed before its limit; the bound is uniform over the allowed modified functions.

theorem qualitative_strong_rough_shifts (hM : PrimeReciprocalInput)
    (hMRT : MRTShortExponentialInput) {f : ℕ → ℂ}
    (hfnp : UniformlyNonpretentious f) (hf : OneBounded f)
    (h : ℕ) (hh : 0 < h) (C₀ : ℝ) (hC₀ : 1 ≤ C₀) :
    ∃ C : ℝ, 0 < C ∧ ∀ᶠ B : ℝ in atTop,
      ∀ (P : Finset ℕ) (D : ℕ),
      (1 / 2 : ℝ) * Real.exp (B ^ (9999 / 10000 : ℝ)) ≤ D →
      (D : ℝ) ≤ Real.exp (C₀ * B ^ (2 : ℕ)) →
      ∀ᶠ Y : ℕ in atTop, ∀ b g : ℕ → ℂ,
      Multiplicative b → OneBounded b → OneBounded g →
      (∀ p, Nat.Prime p → p ∉ P → b p = f p) →
      ∀ (l : ℕ) [NeZero l] (a : ZMod l) (Z : Finset ℕ) (c : ℕ → ℂ),
…

Mentions: OAI.TwoPointCorrelations.PrimeReciprocalInput, OAI.TwoPointCorrelations.MRTShortExponentialInput, OAI.TwoPointCorrelations.UniformlyNonpretentious, OAI.TwoPointCorrelations.OneBounded, OAI.TwoPointCorrelations.Multiplicative