QualitativeRightShifts.lean`

3 declarations, 0 contain sorry, 0 declared as axioms

The qualitative rough-shift estimate when the original second factor is uniformly nonpretentious.

lemma OAI.TwoPointCorrelations.progressionSequence_mod_one line 14
lemma OAI.TwoPointCorrelations.long_window_exp_bound line 19

theorem OAI.TwoPointCorrelations.qualitative_rough_shifts_right line 34

Source documentation

The second-factor counterpart of qualitative_rough_shifts, with the same exponent and uniformity in finite prime changes.

theorem qualitative_rough_shifts_right (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) →
      ∀ᶠ Y : ℕ in atTop, ∀ b g : ℕ → ℂ,
      OneBounded b → Multiplicative g → OneBounded g →
      (∀ p, Nat.Prime p → p ∉ P → g 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